Classification of Lipschitz Derivatives in Terms of Semicontinuity and the Baire Limit Functions

Oleksandr V. Maslyuchenko; Ziemowit M. Wójcicki

doi:10.2478/amsil-2026-0003

1. Introduction

The Lipschitz derivatives are useful tools for investigation of different notions of differentiability. For example, the big Lipschitz derivative Lip f of a given function f often occurs in theorems of Rademacher–Stepanov type (see for example [11, 14, 9, 8, 5]). Statements of this type usually involves the set L(f) = {x ∈ ℝ: Lip f(x) < ∞}. The local Lipschitz derivative 𝕃ip f together with Lip f characterize the local and pointwise Lipschitzness of functions defined on a metric space [7]. The little Lipschitz derivative was introduced by Cheeger in [4] and together with Lip f play important role in the research of the first order differential calculus in metric spaces. The crucial fact is the purely metric character of the definitions of Lipschitz derivatives. The above considerations lead to the natural question of characterizing the sets ℓ(f), L(f) and 𝕃(f) for functions defined on metric spaces. In recent years, this problem has been investigated in many articles, such as [12, 3, 6]. In particular, in [3] it was shown that ℓ(f) is a G_δσ-set and L(f) is an F_σ-set for a function f : ℝ → ℝ. In our approach, we introduce generalized notions of semicontinuity of a function and classify the Lipschitz derivatives with help of these properties. This allows us to easily derive the Borel type of sets ℓ(f), L(f), 𝕃(f) of a function f acting between arbitrary metric spaces.

Another interesting question to consider is whether for a given triplet of functions (u, v, w), there exists a continuous function f such that lip f = u, Lip f = v and 𝕃ip f = w. Some advances in this direction were made in [1] and [2]. Our characterization of Lipschitz derivatives in terms of generalized semicontinuity constrains the possible choice of functions (u, v, w). Moreover, we obtain another necessary criterion for a triple (u, v, w) defined on locally convex subset of a normed space: the upper Baire limits functions of u and v are equal to w.

2. Lipschitz derivatives

Let X be a metric space, a ∈ X and ε > 0. We always denote the metric on X by | · − · |_X and

\begin{array}{l} B (a, ε) = B_{X} (a, ε) = {x \in X : | x - a |_{X} < ε}, \\ B a, ε] = B_{X} a, ε] = {x \in X : | x - a |_{X} \leq ε} . \end{array}

Definition 1.

Let X and Y be metric spaces, f : X → Y be a function, x ∈ X. Denote

$‖ f ‖_{lip} = sup_{u \neq v \in X} \frac{1}{{u - v|}_{X}} | f (u) - f (v) |_{Y}$ ,
$𝕃ip f (x) = \underset{u \neq v}{\underset{(u, v) \to (x, x)}{lim sup}} \frac{1}{{u - v|}_{X}} {f (u) - f (v)|}_{Y}$ ,
$Lip f (x) = \underset{u \to x}{lim sup} \frac{1}{{u - x|}_{X}} | f (u) - f (x) |_{Y}$ ,
$lip f (x) = \underset{r \to 0^{+}}{lim inf} sup_{u \in B (x, r)} \frac{1}{r} | f (u) - f (x) |_{Y}$ ;

The number ∥f∥_lip is the Lipschitz constant of f. The functions 𝕃ip f, Lip f and lip f are called the local, big and little Lipschitz derivative respectively.

We denote by X^d the set of all non-isolated points of X. Throughout the paper, we assume that sup ∅ = 0. As a consequence of this assumption we have 𝕃ip f(x) = Lip f(x) = lip f(x) = 0 for any x ∈ X \ X^d.

Obviously, if Y is a normed space then ∥ · ∥_lip is an extended seminorm on Y^X in the sense [13]. Moreover, ∥ · ∥_lip is a norm on the space Lip_a(X, Y ) of all Lipschitz functions f : X → Y vanishing at some fixed point a ∈ X.

We introduce some auxiliary notations:

${𝕃ip}^{r} f (x) = {f | B_{(x, r)}‖}_{lip} = sup_{u \neq v \in B (x, r)} \frac{1}{{u - v|}_{X}} | f (u) - f (v) |_{Y}$ ;
${Lip}^{r} f (x) = sup_{u \in B (x, r)} \frac{1}{r} | f (u) - f (x) |_{Y}, {Lip}_{+}^{r} f (x) = sup_{u \in B x, r]} \frac{1}{r} | f (u) - f (x) |_{Y}$ ;
${Lip}_{r} f (x) = sup_{0 < ϱ < r} {Lip}^{ϱ} f (x), 1 {ip}_{r} f (x) = inf_{0 < ϱ < r} {Lip}^{ϱ} f (x)$ .

Therefore, the definitions of the Lipschitz derivatives might be rewritten as follows

𝕃ip f (x) = inf_{r > 0} {𝕃ip}^{r} f (x), lip f (x) = \underset{r \to 0^{+}}{lim inf} {Lip}^{r} f (x) .

Some authors (see, for example, [6, 2, 3]) define Lip f and lip f using the function ${Lip}_{+}^{r} f$ instead of Lip^r f. In the case where X is a normed space, we have $B x, r] = \bar{B (x, r)}$ . Therefore, ${Lip}^{r} f (x) = {Lip}_{+}^{r} f (x)$ for any continuous function f. But the previous equality does not hold for the discrete metric on X, nonconstant f and r = 1. However, we have the following

Proposition 2.1.

Let X and Y be metric spaces and f : X → Y be a function. Then, for any non-isolated point x ∈ X, the following equalities hold

\begin{array}{l} Lip f (x) = \underset{r \to 0^{+}}{lim sup} {Lip}^{r} f (x) = \underset{r \to 0^{+}}{lim sup} {Lip}_{+}^{r} f (x), \\ lip f (x) = \underset{r \to 0^{+}}{lim inf} {Lip}_{+}^{r} f (x) . \end{array}

We start with the following

Lemma 2.2.

Let φ, ψ : [0, +∞) → [0, +∞] be functions such that

φ (ϱ) \leq ψ (ϱ) \leq \frac{r}{ϱ} φ (r)

for any 0 < ϱ < r. Then

\underset{r \to 0^{+}}{lim sup} φ (r) = \underset{r \to 0^{+}}{lim sup} ψ (r), \underset{r \to 0^{+}}{lim inf} φ (r) = \underset{r \to 0^{+}}{lim inf} ψ (r) .

Proof.

Denote

\begin{array}{l} A = \underset{r \to 0^{+}}{lim sup} φ (r), B = \underset{r \to 0^{+}}{lim sup} ψ (r), \\ a = \underset{r \to 0^{+}}{lim inf} φ (r), b = lim_{r \to 0^{+}} inf ψ (r) . \end{array}

Obviously, A ≤ B and a ≤ b. Let us show, that A ≥ B. Choose ϱ_n → 0⁺ such that ψ(ϱ_n) → B. Put

r_{n} = ϱ_{n} + ϱ_{n}^{2}

. Since

ψ (ϱ_{n}) \leq \frac{r_{n}}{ϱ_{n}} φ (r_{n})

. Therefore,

A \geq \underset{n \to \infty}{lim sup} φ (r_{n}) \geq lim_{n \to \infty} \frac{ϱ_{n}}{r_{n}} ψ (ϱ_{n}) = B .

Now, we will show that a ≥ b. Choose r_n → 0⁺ such that 0 < r_n < 1 and φ(r_n) → a. Let $ϱ_{n} = r_{n} - r_{n}^{2}$ . Since 0 < ϱ_n < r_n, $ψ (ϱ_{n}) \leq \frac{r_{n}}{ϱ_{n}} φ (r_{n})$ , we have

b \leq \underset{n \to \infty}{lim inf} ψ (ϱ_{n}) \leq lim_{n \to \infty} \frac{r_{n}}{ϱ_{n}} φ (r_{n}) = a .

Proof of Proposition 2.1.

Fix a non-isolated point x ∈ X. Denote φ(r) = Lip^r f(x), $ψ (r) = {Lip}_{+}^{r} f (x)$ . Therefore, $r φ (r) = sup_{u \in B (x, r)} | f (u) - f (x) |_{Y}$ and $r ψ (r) = sup_{u \in B (x, r)} | f (u) - f (x) |_{Y}$ . Let 0 < ϱ < r. Then B (x, ϱ) ⊆ B [x, ϱ] ⊆ B (x, r), so

ϱ φ (ϱ) \leq ϱ ψ (ϱ) \leq r φ (r) .

Hence, φ and ψ satisfy the condition from Lemma 2.2. Thus, by Lemma 2.2 we conclude that

\underset{r \to 0^{+}}{lim sup} φ (r) = \underset{r \to 0^{+}}{lim sup} ψ (r) and \underset{r \to 0^{+}}{lim inf} φ (r) = \underset{r \to 0^{+}}{lim inf} ψ (r) .

By the definition, we have

f (x) = \underset{r \to 0^{+}}{lim inf} φ (r)

It remains to show that $Lip f (x) = \underset{r \to 0^{+}}{lim sup} ψ (r)$ . Denote for any r > 0

α (r) = sup_{0 < ϱ < r} ψ (ϱ) and β (r) = sup \frac{| f (u) - f (x) |_{Y}}{| u - x |_{X}} : 0 < | u - x |_{X} < r\} .

Therefore,

\begin{array}{l} α (r) = sup_{0 < ρ < r} sup_{0 < {u - x|}_{X} \leq ρ} \frac{1}{ρ} | f (u) - f (x) |_{Y} \\ \leq sup_{0 < ρ < r} sup_{0 < {u - x|}_{X} \leq ρ} \frac{1}{{u - x|}_{X}} | f (u) - f (x) |_{Y} = β (r) . \end{array}

On the other hand, we have

\begin{array}{l} β (r) = sup_{0 < ρ < r} sup_{| u - x |_{X} = ρ} \frac{1}{ρ} | f (u) - f (x) |_{Y} \\ \leq sup_{0 < ρ < r} sup_{0 < {u - x|}_{X} \leq ρ} \frac{1}{ρ} | f (u) - f (x) |_{Y} \\ = sup_{0 < ρ < r} {Lip}_{+}^{ρ} f (x) = α (r) . \end{array}

We have shown, that α(r) = β(r) for r > 0. Hence,

Lip f (x) = inf_{r > 0} β (r) = inf_{r > 0} α (r) = \underset{r \to 0^{+}}{lim sup} ψ (r) .

Note, that

(2.1)

{Lip}_{r} f (x) \leq {Lip}_{r^{'}} f (x) and {lip}_{r} f (x) \geq {lip}_{r^{'}} f (x) if 0 < r < r^{'} .

So, the definitions and the previous proposition yield

(2.2)

Lip f (x) = inf_{r > 0} {Lip}_{r} f (x) = lim_{r \to 0^{+}} {Lip}_{r} f (x),

(2.3)

lip f (x) = sup_{r > 0} 1 {ip}_{r} f (x) = lim_{r \to 0^{+}} 1 {ip}_{r} f (x) .

Therefore, it is easy to see that the following inequalities hold.

(2.4)

\begin{array}{c} {lip}_{r} f (x) \leq {Lip}_{r} f (x) \leq {𝕃ip}^{r} f (x) for any r > 0, \\ lip f (x) \leq Lip f (x) \leq 𝕃ip f (x) . \end{array}

Definition 2.

Let X and Y be metric spaces and γ ≥ 0. A function f : X → Y is called

γ-Lipschitz if ∥f∥_lip ≤ γ;
Lipschitz if ∥f∥_lip < ∞;
locally Lipschitz if 𝕃ip f < ∞;
pointwise Lipschitz if Lip f < ∞;
weakly pointwise Lipschitz if lip f < ∞.

Denote

𝕃(f) = {x ∈ X : 𝕃ip f(x) < ∞};
𝕃^∞(f) ={x ∈ X : 𝕃ip f(x) = ∞} = X \ 𝕃(f);
L(f) = {x ∈ X : Lip f(x) < ∞};
L^∞(f) = {x ∈ X : Lip f(x) = ∞} = X \ L(f);
ℓ(f) = {x ∈ X : lip f(x) < ∞};
ℓ^∞(f) = {x ∈ X : lip f(x) < ∞}; = X \ ℓ(f).

Inequalities ( 2.4) yield the next assertion.

Proposition 2.3.

Let X and Y be metric spaces, and f : X → Y be a function. Then 𝕃(f) ⊆ L(f) ⊆ ℓ(f) and ℓ^∞(f) ⊆ L^∞(f) ⊆ 𝕃^∞(f).

3. Connections of Lipschitz derivatives to classical notion of a derivative

One might ask under what conditions the Lipschitz derivative (of any given type) coincides with one of the “traditional” notions of the derivative of a given function, provided that an appropriate derivative exists. It is obvious that for a real differentiable function f : ℝ → ℝ, we have lip f(x) = Lip f(x) = |f′(x)| at any x ∈ ℝ.

In [7], the following theorem was proved.

Theorem 3.1.

Let X and Y be normed spaces, G be an open subset of X, and f : G → Y have a locally bounded Gateaux derivative f′. Then, $𝕃ip f (x) = \underset{u \to x}{lim sup} ‖ f^{'} (x) ‖$ for x ∈ X and so, f is locally Lipschitz. Moreover, if f is C¹ function, then 𝕃ip f(x) = ∥f′(x)∥, x ∈ X.

The following result was also stated in [7], but the proof contains a small blunder. Here, we provide the correct proof.

Theorem 3.2.

Let f : X → Y, where X and Y are normed spaces and assume that there exists the Fréchet derivative df (x₀) of f at a point x₀ ∈ X. Then lip f(x₀) = Lip f(x₀) = ∥df (x₀)∥.

Proof.

It is enough to consider the case X ≠ {0}. Denote by A = df (x₀) the Fréchet derivative of f at the point x₀. We have

(3.1)

f (x) - f (x_{0}) = A (x - x_{0}) + α (x) for x \in X,

where α is a function, such that

lim_{x \to x_{0}} \frac{α (x)}{‖ x - x_{0} ‖} = 0

. By (3.1) we have

‖ f (x) - f (x_{0}) ‖ \leq ‖ A ‖ ‖ x - x_{0} ‖ + ‖ α (x) ‖,

hence

\frac{‖ f (x) - f (x_{0}) ‖}{| | x - x_{0} ‖} \leq ‖ A ‖ + ‖ \frac{α (x)}{‖ x - x_{0} ‖} ‖ .

Thus,

\begin{array}{l} lip f (x_{0}) \leq Lip f (x_{0}) = \underset{x \to x_{0}}{lim sup} \frac{‖ f (x) - f (x_{0}) ‖}{| | x - x_{0} ‖} \\ \leq lim_{x \to x_{0}} (‖ A ‖ + \frac{α (x)}{‖ x - x_{0} ‖}‖) = ‖ A ‖ . \end{array}

We want to prove that lip f(x₀) ≥ ∥A∥. Fix ε > 0. Then, there exists e ∈ X such that ∥e∥ = 1 and

A e‖ \geq A‖ - ε .

For any r > 0, denote x_r = x₀ + re. Note that r = ∥x_t − x₀∥, and x_r → x₀ as r → 0. So,

\frac{α (x_{r})}{r} \to 0

as r → 0. Therefore, by (3.1)

f (x_{r}) - f (x_{0}) = A (x_{r} - x_{0}) + α (x_{r}) = r A e + α (x_{r}),

so for any r > 0

\begin{array}{l} {Lip}_{+}^{r} f (x_{0}) = \frac{1}{r} sup_{‖ u - x ‖ \leq r} ‖ f (x) - f (x_{0}) ‖ \geq \frac{1}{r} ‖ f (x_{r}) - f (x_{0}) ‖ \\ = A e + \frac{α (x_{r})}{r}‖ \geq ‖ A e ‖ - \frac{α (x_{r})}{r}‖ \geq ‖ A ‖ - ε - \frac{α (x_{r})}{r}‖ . \end{array}

Thus, by Proposition 2.1 we conclude that

lip f (x_{0}) = \underset{r \to 0^{+}}{lim inf} {Lip}_{+}^{r} f (x_{0}) \geq lim_{r \to 0^{+}} (‖ A ‖ - ε - \frac{α (x_{r})}{r}‖) = ‖ A ‖ - ε .

Since the ε was chosen arbitrarily, the proof is finished.

4. Semicontinuity with respect to a family of sets

In this section we introduce some modification of semicontinuity, which will help us to classify the Lipschitz derivatives.

Let X be a topological space and 𝒜 be a family of subsets of X. We denote

\begin{array}{l} A_{c} = X \ A : A \in A\}, \\ A_{σ} = \cup_{n = 1}^{\infty} A_{n} : A_{n} \in A for all n \in 𝕅\}, \\ A_{δ} = \cap_{n = 1}^{\infty} A_{n} : A_{n} \in A for all n \in 𝕅\} . \end{array}

We will also combine these symbols. It is easy to check, for example, that 𝒜_cδc = 𝒜_σ, 𝒜_σc = 𝒜_cδ, 𝒜_δc = 𝒜_cσ and so on. If 𝒯 denotes the topology of X, then applying above notation to the family 𝒜 = 𝒯, the 𝒯_δ is the familiar Borel class 𝒢_δ of G_δ-subsets of X. Complementary, the family 𝒯_cσ is the Borel class ℱ_σ of F_σ-subsets of X.

We say that $f : X \to \bar{𝕉}$ is an 𝒜-upper (𝒜-lower) semicontinuous function if f⁻¹ ([−∞, γ)) ∈ 𝒜 (resp. f⁻¹ ((γ, +∞]) ∈ 𝒜) for any γ ∈ ℝ. If 𝒜 = 𝒯 is the topology of X, then we omit the symbol 𝒜 in the previous definitions. For our purposes, the ℱ_σ-upper and lower semicontinuous functions are particularly important.

Proposition 4.1.

Let X be a topological space, 𝒜 ⊆ 2^X and $f : X \to \bar{𝕉}$ be an A-upper semicontinuous function. Then

(i) f⁻¹ ([γ, +∞]) ∈ 𝒜_c for any γ ∈ ℝ;
(ii) f⁻¹ [[−∞, +∞)] ∈ 𝒜_σ and, so, f⁻¹ [{+∞}] ∈𝒜_σc;
(iii) f⁻¹ [{−∞}] ∈ 𝒜_δ and, so, f⁻¹ ((−∞, +∞]) ∈𝒜_δc;
(iv) f is 𝒜_cσ-lower semicontinuous.

Proof.

(i) For any γ ∈ ℝ we have that f⁻¹ ([−∞, γ)) ∈ 𝒜 and then
$f^{- 1} (γ, + \infty]) = X \ f^{- 1} (- \infty, γ)) \in A_{c} .$
(ii) Since f⁻¹ [[−∞, n)] ∈ 𝒜 for any n ∈ ℕ, we conclude that
$f^{- 1} - \infty, + \infty)] = \cup_{n = 1}^{\infty} f^{- 1} - \infty, n)] \in A_{σ},$
and so, f⁻¹ [{+∞}] = X \ f⁻¹ [[−∞, +∞)] ∈ 𝒜_{σ c}.
(iii) Since f⁻¹ [[−∞, −n)] ∈ 𝒜 for any n ∈ ℕ, we have that
$f^{- 1} - \infty\}] = \cap_{n = 1}^{\infty} f^{- 1} - \infty, - n)] \in A_{δ},$
and so, f⁻¹ [(−∞, +∞]] = X \ f⁻¹ [{−∞}]∈ 𝒜_δc.
(iv) Let γ ∈ ℝ and γ_n ↓ γ. Since f⁻¹ [[γ_n, +∞]] ∈ 𝒜_c by (i), we conclude that
$f^{- 1} γ, + \infty]] = \cup_{n = 1}^{\infty} f^{- 1} γ_{n}, + \infty]] \in A_{c σ},$
i.e., f is 𝒜_cσ-lower semicontinuous.

Proposition 4.2.

Let X be a topological space, 𝒜 ⊆ 2^X, $f_{x} : X \to \bar{𝕉}$ , be an 𝒜-upper semicontinuous function for any n ∈ ℕ and $f : X \to \bar{𝕉}$ be a function such that f(x) = sup_n_∈ℕ f_n(x) for any x ∈ X. Then f is an 𝒜_cσ-lower semicontinuous function.

Proof.

Consider γ ∈ ℝ. By Proposition 4.1(iv) the functions f_n are 𝒜_cσ-lower semicontinuous. So, $f_{n}^{- 1} γ, + \infty]] \in A_{c σ}$ for any n ∈ ℕ. Consequently,

f^{- 1} γ, + \infty]] = \cup_{n = 1}^{\infty} f_{n}^{- 1} γ, + \infty]] \in A_{c σ} .

Thus, f is an 𝒜_cσ-lower semicontinuous function.

Observe that f is an 𝒜-upper semicontinuous function if and only if −f is 𝒜-lower semicontinuous. Therefore, using Proposition 4.1 and 4.2 with g = −f we obtain the following two propositions.

Proposition 4.3.

Let X be a topological space, 𝒜 ⊆ 2^X and $f : X \to \bar{𝕉}$ be an 𝒜-lower semicontinuous function. Then

(i) f⁻¹ [[−∞, γ]] ∈𝒜_c for any γ ∈ ℝ;
(ii) f⁻¹ [(−∞, +∞]] ∈ 𝒜_σ and, so, f⁻¹ [{−∞}] ∈ 𝒜_{σ c};
(iii) f⁻¹ [{+∞}] ∈ 𝒜_δ and, so, f⁻¹ [[−∞, +∞)] ∈ 𝒜_δc;
(iv) f is 𝒜_cσ-upper semicontinuous.

Proposition 4.4.

Let X be a topological space, 𝒜 ⊆ 2^X, $f_{n} : X \to \bar{𝕉}$ be an 𝒜-lower semicontinuous function for any n ∈ℕ and $f : X \to \bar{𝕉}$ be a function such that f(x) = inf_n_∈ℕf_n(x) for any x ∈ X. Then f is an 𝒜_cσ-upper semicontinuous function.

5. Classification of the Lipschitz derivatives

Now we pass to the investigation of the type of semicontinuity of Lipschitz derivatives of continuous functions. In [3] semicontinuity of Lipschitz derivatives of a continuous function f : ℝ → ℝ was obtained from the continuity of Lip^r f. But in the general situation, this function need not be continuous. Therefore, we prove semicontinuity of Lipschitz derivatives directly from the definitions.

Lemma 5.1.

Let X and Y be metric spaces, f : X → Y be a continuous function and r > 0. Then lip_r f : X → [0, +∞] is an upper semicontinuous function.

Proof.

Let x₀ ∈ X and γ > lip_r f(x₀). Then

inf_{ϱ < r} {Lip}^{ϱ} f (x_{0}) = {lip}_{r} f (x_{0}) < γ .

So, there is positive ϱ < r such that Lip^ϱ f(x₀) < γ. Pick γ₁ such that Lip^ϱ f(x₀) < γ₁ < γ. Then we choose ϱ₁ such that

\frac{γ_{1}}{γ} ϱ < ϱ_{1} < ϱ

. So, γϱ₁ > γ₁ϱ. Therefore,

sup_{u \in B (x_{0}, ϱ)} | f (u) - f (x_{0}) |_{Y} = ϱ {Lip}^{ϱ} f (x_{0}) < γ_{1} ϱ .

Then

| f (u) - f (x_{0}) |_{Y} < γ_{1} ϱ for any u \in B (x_{0}, ϱ) .

By the continuity of f at x₀ there exists δ> 0 such that ϱ₁ + δ < ϱ and

| f (x) - f (x_{0}) |_{Y} < γ ϱ_{1} - γ_{1} ϱ for any x \in U = B (x_{0}, δ) .

Consider x ∈ U and u ∈ B(x, ϱ₁). Then

u - x_{0} |_{X} \leq| u - x |_{X} + | x - x_{0} |_{X} < ϱ_{1} + δ < ϱ,

and so, u ∈ B(x₀, ϱ). Therefore,

f (u) - f (x) |_{Y} \leq| f (u) - f (x_{0}) |_{Y} + | f (x_{0}) - f (x) |_{Y} < γ_{1} ϱ + (γ ϱ_{1} - γ_{1} ϱ) = γ ϱ_{1} .

Thus,

\frac{1}{ϱ_{1}} | f (u) - f (x) |_{Y} \leq γ

for any u ∈ B(x, ϱ₁). Hence, Lip^ϱ¹ f(x) ≤ γ. But 0 < ϱ₁ < r. Therefore, lip_r f(x) ≤ γ for any x ∈ U. Thus, lip_r f is upper semicontinuous at x₀.

Theorem 5.2.

Let X and Y be metric spaces and f : X → Y be a continuous function. Then lip f : X → [0, +∞] is a ℱ_σ-lower semicontinuous function.

Proof.

By (2.1) and (2.3) we conclude that $lip f (x) = {sup}_{n \in 𝕅} {lip}_{\frac{1}{n}} f (x)$ for any x ∈ X. By Lemma 5.1, the functions ${lip}_{\frac{1}{n}} f$ are 𝒯-upper semicontinuous, where 𝒯 is the topology of X. Therefore, by Proposition 4.2 lip f is 𝒯_cσ-lower semicontinuous. This means that lip f is ℱ_σ-lower semicontinuous.

Lemma 5.3.

Let X and Y be metric spaces, f : X → Y be a continuous function and r > 0. Then Lip_r f : X → [0, +∞] is a lower semicontinuous function.

Proof.

Fix r > 0. Let x₀ ∈ X and γ < Lip_r f(x₀). Then

sup_{ϱ < r} {Lip}^{ϱ} f (x_{0}) = {Lip}_{r} f (x_{0}) > γ .

So, there is ϱ ∈ (0, r) such that Lip^ϱ f(x₀) > γ. Pick γ₁ such that

γ < γ_{1} < {Lip}^{ϱ} f (x_{0}) .

Therefore,

sup_{u \in B (x_{0}, ϱ)} {f (u) - f (x_{0})|}_{Y} = ϱ {Lip}^{ϱ} f (x_{0}) > γ_{1} ϱ .

Thus, there is u ∈ B(x₀, ϱ) with

{f (u) - f (x_{0})|}_{Y} > γ_{1} ϱ .

Then we choose ϱ₁ such that

ϱ < ϱ_{1} < min r, \frac{γ_{1}}{γ} ϱ\}

. Consequently, γϱ₁ < γ₁ϱ. By the continuity of f at x₀ there exists δ > 0 such that ϱ + δ < ϱ₁ and

| f (x) - f (x_{0}) |_{Y} < γ_{1} ϱ - γ ϱ_{1} for any x \in U : = B (x_{0}, δ) .

Consider x ∈ U. Then

u - x |_{X} \leq| u - x_{0} |_{X} + | x_{0} - x |_{X} < ϱ + δ < ϱ_{1},

and, so, u ∈ B(x, ϱ₁). Consequently,

f (u) - f {(x)}_{Y}| \geq {f (u) - f (x_{0})|}_{Y} - {f (x) - f (x_{0})|}_{Y} > γ_{1} ϱ - (γ_{1} ϱ - γ ϱ_{1}) = γ ϱ_{1} .

Hence, Lip^ϱ¹ f(x) > γ. But 0 < ϱ₁ < r. Therefore, Lip_r f(x) > γ for any x ∈ U. Thus, Lip_r f is lower semicontinuous at x₀.

Theorem 5.4.

Let X and Y be metric spaces and f : X → Y be a continuous function. Then Lip f : X → [0, +∞] is a ℱ_σ-upper semicontinuous function.

Proof.

By (2.1) and (2.2) we conclude that $Lip f (x) = {inf}_{n \in 𝕅} {Lip}_{\frac{1}{n}} f (x)$ for any x ∈ X. By Lemma 5.3, the functions ${Lip}_{\frac{1}{n}} f$ are 𝒯-lower semicontinuous where 𝒯 is the topology of X. Therefore, by Proposition 4.4 Lip f is 𝒯_cσ-upper semicontinuous. This means that Lip f is ℱ_σ-upper semicontinuous.

Theorem 5.5.

Let X and Y be metric spaces and f : X → Y be a function. Then 𝕃ip f : X → [0, +∞] is an upper semicontinuous function.

Proof.

Fix x₀ ∈ X and γ > 𝕃ip f(x₀). Since $𝕃ip f (x_{0}) = inf_{r > 0} {𝕃ip}^{r} f (x_{0})$ , there exists r > 0 such that 𝕃ip^r f(x₀) < γ. Set $ϱ = \frac{r}{2}$ and consider x ∈ B(x₀, ϱ). Then B(x, ϱ) ⊆ B(x₀, r). Consequently,

\begin{array}{l} 𝕃ip f (x) \leq {𝕃ip}^{ϱ} f (x) = sup_{u \neq v \in B (x, ϱ)} \frac{1}{{u - v|}_{X}} | f (u) - f (v) |_{Y} \\ \leq sup_{u \neq v \in B (x_{0}, r)} \frac{1}{{u - v|}_{X}} | f (u) - f (v) |_{Y} = {𝕃ip}^{r} f (x_{0}) < γ \end{array}

and, hence, 𝕃ip f is upper semicontinuous.

Theorems 5.2, 5.4, 5.5, and Propositions 4.1, 4.3 yield the following assertions.

Corollary 5.6.

Let X and Y be metric spaces and f : X → Y be a continuous function. Then

(i) ℓ(f) is a G_δσ-set;
(ii) ℓ^∞(f) is an F_σδ-set;
(iii) L(f) is an F_σ-set;
(iv) L^∞(f) is a G_δ-set.

Proof.

(i) We have
$\begin{array}{l} 𝓁 (f) = {x \in X : lip f (x) < \infty} = {(lip f)}^{- 1} [[- \infty, + \infty)] \\ = X \ {(lip f)}^{- 1} [{+ \infty}] \end{array}$
and, since lip f is ℱ_σ-lower semicontinuous, by Proposition 4.3(iii)
${(lip f)}^{- 1} + \infty\}] \in F_{σ δ},$
so ℓ(f) = X \ (lip f)⁻¹ [{+∞}] ∈ 𝒢_{δ σ}.
(ii) It follows immediately from (i).
(iii) It is easy to see, that $L (f) = \cup_{k = 1}^{\infty} {(Lip f)}^{- 1} [[0, k)]$ . Since Lip f is an ℱ_σ-upper semicontinuous function, each set (Lip f)⁻¹[[0, k)] is of F_σ type, hence L(f) is an F_σ-set as a countable sum of F_σ-sets.
(iv) It follows from (iii).

Corollary 5.7.

Let X and Y be metric spaces and f : X → Y be an arbitrary function. Then

(i) 𝕃(f) is an open set;
(ii) 𝕃^∞(f) is a closed set.

6. Characterization of Lipschitz functions on a convex subset of a normed space

The following lemma was applied by Buczolich, Hanson, Maga and Vértesy in certain investigations of Lipschitz derivatives of the real functions of real variable.

Lemma 6.1 ([2, Lemma 2.2])

If E ⊆ ℝ and f : ℝ → ℝ such that lip f ≤ χ_E then |f(a) − f(b)| ≤ µ([a, b] ∩ E) for every a, b ∈ ℝ (where a < b) so, f is Lipschitz and hence absolutely continuous.

In the above, µ denotes the Lebesgue measure. We will state the following

Corollary 6.2.

Let γ > 0 and f : [0, 1] → ℝ be a function such that lip f(x) ≤ γ for any x ∈ [0, 1]. Then f is γ-Lipschitz.

Proof.

Extend f to $\tilde{f} : 𝕉 \to 𝕉$ by $\tilde{f} (x) = f (0)$ if x < 0 and $\tilde{f} (x) = f (1)$ if x > 1. Let $g = \frac{1}{γ} \tilde{f}$ and E = [0, 1]. Then by Lemma 6.1 we conclude that $\frac{1}{γ} f (x) - f (y)| = g (x) - g (y)| \leq μ (x, y] \cap E) = x - y|$ for any x, y ∈ [0, 1].

The next result will allow us to apply Lemma 6.1 and Corollary 6.2 for functions defined on normed spaces.

Lemma 6.3.

Let A be a convex subset of the normed space X, f : X → ℝ be a function, and a, b ∈ A. Moreover, let T : [0, 1] → A be an affine function given by T (u) = a + u(b − a) for 0 ≤ u ≤ 1 and g = f ◦ T : [0, 1] → ℝ. Then,

(6.1)

lip g \leq ‖ b - a ‖ ((lip f) \circ T) .

Proof.

It is enough to consider the case where a ≠ b. Fix u₀∈ [0, 1] and observe, that

(6.2)

‖ T (u) - T (u_{0}) ‖ = ‖ (u - u_{0}) (b - a) ‖ = u - u_{0}| ‖ b - a ‖, 0 \leq u \leq 1.

We have

(6.3)

lip g (u_{0}) = \underset{r \to 0^{+}}{lim inf} sup_{0 < u - u_{0}| < r} \frac{f (T (u)) - f (T (u_{0}))|}{r} .

Put x₀ = T (u₀). Substituting ϱ = r ∥b − a∥ and x = T (u) in (6.3) and taking into account (6.2) we obtain that

\begin{array}{l} lip g (u_{0}) = \underset{ϱ \to 0^{+}}{lim inf} sup_{0 < u - u_{0}| < \frac{ϱ}{‖ b - a ‖}} \frac{f (T (u)) - f (T (u_{0}))|}{ϱ / ‖ b - a ‖} \\ = ‖ b - a ‖ \underset{ϱ \to 0^{+}}{lim inf} \underset{x \in T 0, 1]]}{sup_{0 < ‖ x - x_{0} ‖ < ϱ}} \frac{f (x) - f (x_{0})|}{ϱ} \\ \leq ‖ b - a ‖ \underset{ϱ \to 0 +}{lim inf} sup_{0 < ‖ x - x_{0} ‖ < ϱ} \frac{f (x) - f (x_{0})|}{ϱ} \\ = ‖ b - a ‖ lip f (x_{0}) . \end{array}

((lip f) \circ T) ‖ b - a ‖ = ((lip f) \circ T) \cdot \frac{d T}{d u}‖

, the right side of inequality (6.1) is reminiscent of the “chain rule” for the usual derivative. Nevertheless, the inequality can be strict. To see that, take a function f : ℝ² → ℝ defined as f(x, y) = y, (x, y) ∈ ℝ² and consider the usual distance on ℝ². By Theorem 3.2 we have,

lip f (x, y) = ‖ d f (x, y) ‖ = \sqrt{{(\frac{\partial f}{\partial x} (x, y))}^{2} + {(\frac{\partial f}{\partial y} (x, y))}^{2}} = 1

. Let a = (0, 0), b = (1, 0), T (u) = a + (b − a)u = (u, 0), u ∈ [0, 1] and g = f ◦ T. Therefore, g(u) = f(u, 0) = 0 and so, lip g(u) = 0 < 1 = ∥b − a∥ lip f(T (u)) for any u.

Theorem 6.4.

Let D be a convex subset of a normed space X, f : D → ℝ be a function and γ ≥ 0. Then f is γ-Lipschitz if and only if lip f(x) ≤ γ for any x ∈ D.

Proof.

Fix a, b ∈ D. Define T : [0, 1] → D as

T (u) = a + u (b - a) for u \in 0, 1]

and put g = f ◦ T : [0, 1] → ℝ. Applying Lemma 6.3 we get

lip g (u) \leq ‖ b - a ‖ lip f (T (u)) \leq ‖ b - a ‖ γ,

for any u ∈ [0, 1]. Therefore, Corollary 6.2 implies that g is Lipschitz with the constant γ₁ = ∥b − a∥ γ. Thus,

f (a) - f (b)| = g (0) - g (1)| \leq γ_{1} 0 - 1| = γ ‖ a - b ‖ .

So, f is γ-Lipschitz on D.

By ∥·∥_∞ we denote standard norm on space of bounded real functions B(D) defined on a set D, i.e., $‖ h ‖_{\infty} = sup_{x \in D} h (x)|$ for any h: D → ℝ.

Corollary 6.5.

Let f : D → ℝ be a continuous function, where D is a convex subset of some normed space X. Then, ∥f∥_lip = ∥lip f∥_∞.

Proof.

We simply check, that if ∥f∥_lip < ∞ or ∥lip f∥_∞ < ∞, then

\begin{array}{l} {f‖}_{lip} = inf γ > 0 : f is γ - Lipschitz\} \\ = inf γ > 0 : lip f (x) \leq γ for any x \in D\} \\ = sup_{} lip f (x) = ‖ lip f ‖_{\infty}, \end{array}

where the second equality follows from Theorem 6.4.

Therefore, lip is an isometric injection of the normed space Lip_a(D, ℝ) with some a ∈ X, into the space B(D).

7. Baire limit functions of Lipschitz derivatives

For a given function $f : X \to \bar{𝕉}$ , defined on a metric space X, its upper Baire function f^∨ is defined by

f^{\lor} (x) = inf_{U \in U (x)} sup_{u \in U} f (u), x \in X,

and its lower Baire function f^∧ is defined by

f^{\land} (x) = sup_{U \in U (x)} inf_{u \in U} f (u), x \in X,

where 𝒰 (x) is the family of all the neighborhoods of x in X. (See, for example, [10].) The upper Baire function f^∨ is upper semicontinuous and the lower Baire function f^∧ is lower semicontinuous.

A subset D of a normed space X is called locally convex if for any point x ∈ D and any neighborhood U of x in D there is a convex neighborhood V of x in D such that V ⊆ U. For example, every convex set and every open set in X is locally convex.

Theorem 7.1.

Let D be a locally convex subset of a normed space X and let f : D → ℝ be a function. Then

{(lip f)}^{\lor} = {(Lip f)}^{\lor} = 𝕃ip f .

Proof.

Since lip f ≤ Lip f ≤ 𝕃ip f and 𝕃ip f is upper semicontinuous by Theorem 5.5, we have

{(lip f)}^{\lor} \leq {(Lip f)}^{\lor} \leq {(𝕃ip f)}^{\lor} = 𝕃ip f .

Therefore, it is enough to prove that 𝕃ip f ≤ (lip f)^∨. Fix x₀ ∈ D. The case where (lip f)^∨(x₀) = ∞ is obvious. So, we suppose that (lip f)^∨(x₀) < ∞. Let γ > (lip f)^∨(x₀). Then, there exists a convex neighborhood U of x₀, such that

lip f (x) < γ for any x \in U .

By Theorem 6.4, the function f is γ-Lipschitz on U. Hence,

\begin{array}{l} 𝕃ip f (x_{0}) = inf_{r > 0} {f |_{B (x_{0}, r)}‖}_{lip} \\ \leq ‖ f |_{U} ‖_{lip} \leq γ, \end{array}

where B (x₀, r) means the ball in the metric subspace D. Passing to the limit with γ → (lip f)^∨(x₀) we obtain the desired inequality.

Acknowledgments.

The authors would like to appreciate the referee for his/her many helpful comments and suggestions throughout this paper.

Classification of Lipschitz Derivatives in Terms of Semicontinuity and the Baire Limit Functions

Full Article

1. Introduction

2. Lipschitz derivatives

Definition 1.

Proposition 2.1.

Lemma 2.2.

Proof.

Proof of Proposition 2.1.

(2.1)

(2.2)

(2.3)

(2.4)

Definition 2.

Proposition 2.3.

3. Connections of Lipschitz derivatives to classical notion of a derivative

Theorem 3.1.

Theorem 3.2.

Proof.

(3.1)

4. Semicontinuity with respect to a family of sets

Proposition 4.1.

Proof.

Proposition 4.2.

Proof.

Proposition 4.3.

Proposition 4.4.

5. Classification of the Lipschitz derivatives

Lemma 5.1.

Proof.

Theorem 5.2.

Proof.

Lemma 5.3.

Proof.

Theorem 5.4.

Proof.

Theorem 5.5.

Proof.

Corollary 5.6.

Proof.

Corollary 5.7.

6. Characterization of Lipschitz functions on a convex subset of a normed space

Lemma 6.1 ([2, Lemma 2.2])

Corollary 6.2.

Proof.

Lemma 6.3.

(6.1)

Proof.

(6.2)

(6.3)

Theorem 6.4.

Proof.

Corollary 6.5.

Proof.

7. Baire limit functions of Lipschitz derivatives

Theorem 7.1.

Proof.

Acknowledgments.