I want to prove that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is a $C^1$ diffeomorphism then $f$ restricted to a compact subset $K$ of $\mathbb{R}^n$ is uniformly differentiable, that is, for every $\varepsilon>0$, there exists $\delta>0$, such that $|h|<\delta$ implies $$|f(x+h)−f(x)−Df(x).h|<\varepsilon|h|$$ for every $x\in K$ (the number $\delta$ does not depend on $x$).
It seems that the $C^1$ hypothesis is essential, but even in this case I am not being able to prove this. Also, do you know any example of a diffeomorphism, defined in a compact subset, that is not uniformly differentiable?
Let $f \colon \mathbb{R}^n \to \mathbb{R}^m$ be a $C^1$ function (notice that $n \ne m$ is not excluded). Let $K \subset \mathbb{R}^n$. For any $\epsilon > 0$ there exists $\delta > 0$ such that if $x \in K$ and $\lVert h \rVert < \delta$ then $\lVert Df(x) - Df(x + h) \rVert < \epsilon$.
Take $\epsilon > 0$, $x \in K$ and $h$ with $\lVert h \rVert < \delta$. By the $C^1$ Mean Value Theorem (see, e.g., Thm. 12 on p. 278 of: C. C. Pugh, Real Mathematical Analysis), there holds $$ f(x + h) - f(x) = \Bigl(\int\limits_{0}^{1} Df(x + t h) \, dt \Bigr) h. $$ Therefore, $$ f(x + h) - f(x) - Df(x) h = \Bigl(\int\limits_{0}^{1} (Df(x + t h) - Df(x)) \, dt \Bigr) h $$ with $$ \Bigl\lVert \int\limits_{0}^{1} (Df(x + t h) - Df(x)) \, dt \Bigr\rVert \le \int\limits_{0}^{1} \lVert Df(x + t h) - Df(x) \rVert \, dt < \epsilon. $$