Lipschitz Estimate for $C^1$ Functions*: Let $U \subseteq \mathbb{R}^n$ be open and $F: U \to \mathbb{R}^m$ continuously differentiable ($C^1$). Then $F$ is Lipschitz continuous on every compact convex subset of $K \subseteq U$, with Lipschitz constant $\sup\limits_{x \in K} || DF(x) ||$.

Corollary: Continuously differentiable functions $U \to \mathbb{R}^m$ are locally Lipschitz continuous.

This Lemma and its corollary are used in the proofs of the Inverse Function Theorem as well as the Fundamental Theorem for ODE's.

Are there any other (notable) examples of where this Lemma and/or its corollary are applied?

Is it only useful for constructing theoretical bounds for theoretical existence results like the Inverse Function Theorem or the Fundamental Theorem for ODE's? Or can it also give useful numerical bounds (i.e. for applications requiring explicit constructions)?

Context: I am trying to gauge how important this lemma is "by itself", e.g. outside of the context of the above two mentioned theorems. First, purely out of curiosity, and second, because I am writing up a proof of the Inverse Function Theorem, and am trying to decide whether this result should be included as "part of the proof" of the Inverse Function Theorem, or treated as an independently important and useful result, e.g. like the Banach Fixed Point Theorem.

*Proposition C.29 and Corollary C.30 in Lee's Introduction to Smooth Manifolds, Theorem 9.19 in Rudin's Principles of Mathematical Analysis.


Assume (for now) that the following is true:

A locally Lipschitz function $f: \mathbb{R} \to X$ is Lipschitz on compact intervals.

Then the local Lipschitz estimate for continuously differentiable functions can be used to conclude the following important theorem from differential geometry:

Continuously differentiable curves are rectifiable.

This follows according to the following claim made here:

Consider a metric space $(X,d)$ and a continuous function $\gamma: [0,1] \to X$.

$\gamma$ is a parametrization of a rectifiable curve if there is a homeomorphism $\varphi:[0,1] \to [0,1]$ such that the map $\gamma \circ \varphi$ is Lipschitz.

In particular, every compact interval is homeomorphic to $[0,1]$, so a Lipschitz function from a compact interval to $X$ defines a rectifiable curve.

Proof that locally Lipschitz implies Lipschitz on compact intervals:

Because $f$ is locally Lipschitz, for every point $x \in [a,b]$ there is a neighborhood $U_x \subset [a,b]$ on which $f$ is Lipschitz. Then the set of all $U_x$ for all $x \in [a,b]$ is an open cover of $[a,b]$. So there is a finite subcover, since $[a,b]$ is compact; i.e. there are finitely many $x_1, \dots, x_n$ such that $f$ is Lipschitz on each $U_{x_i}$ and the $U_{x_i}$ cover all of $[a,b]$. Define $K_i$ for each $i$ to be the Lipschitz constant of $f$ on $U_{x_i}$. Then it follows that $K:=\max_{1 \le i \le n}K_i$ is a Lipschitz constant for $f$ on all of $[a,b]$.

Namely, if $y_1, y_2 \in [a,b]$, then there exists a partition of $[a,b]$ between $y_1$ and $y_2$, $y_1=t_0 < \dots < t_m = y_2$, such that each two consecutive points of the partition are in one of the $U_{x_i}$. So then we have: $$d(f(y_1),f(y_2))\le \sum_{j=1}^m d(f(t_j),f(t_{j-1})) \le \sum_{j=1}^m K_i |t_j - t_{j-1}| \le \sum_{j=1}^m K |t_j - t_{j-1}| = K|y_2 - y_1| \,.$$ Thus $K$ is a Lipschitz constant for $f$ on all of $[a,b]$.

Note that differentiability alone is not enough to guarantee locally Lipschitz, and also that differentiability alone is not enough to guarantee rectifiability, as the example of $f(x)=x \sin(1/x)$ shows. The local Lipschitz estimate for continuous functions explains this similarity.


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