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Is there an example of a function $f \colon [a,b] \to \mathbb{R} $ that satisfies the following conditions:

  1. $f$ is strictly monotone.
  2. $\exists r>0 \colon \forall x \in [a,b], \forall h \in \mathbb{R} \colon |\frac{f(x+h) - f(x)}{h}| < r $

which is not differentiable at some point in $[a,b]$?

Inspired by the non-differentiability of the Cantor function, where at each removed point there is a partial limit of the derivative definition, that goes to $\infty$.

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Yes. Define$$\begin{array}{rccc}f\colon&[-1,1]&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}x&\text{ if }x<0\\2x&\text{otherwise.}\end{cases}\end{array}$$

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