Monotone Function, Derivative Limit Bounded - Differentiable?

Question

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

$f$ is strictly monotone.
$\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 $\infty$.

score 2 · Accepted Answer · answered Apr 20 '18 at 13:24

2

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

answered Apr 20 '18 at 13:24

José Carlos Santos

427,504

will such a function have only a finite number of points of non-differentiability? Do they have to be isolated? Is it possible for them to be dense? – co.sine Apr 20 '18 at 13:30
@co.sine Please post that as another question. – José Carlos Santos Apr 20 '18 at 13:50
For pretty much any question I suspect co.sine could formulate, the answer is probably somewhere in my 3 November 2000 sci.math post ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS. For a stronger notion of "finitely differentiable", see my answer to “Strong” derivative of a monotone function. – Dave L. Renfro Apr 20 '18 at 16:16
@JoséCarlosSantos Posted as another question: https://math.stackexchange.com/questions/2746930/monotone-function-derivative-limit-bounded-differentiable-2?noredirect=1&lq=1 – co.sine Apr 21 '18 at 07:06
@DaveL.Renfro thank you, I'll have a look at your post. Will you also look at my question? -https://math.stackexchange.com/questions/2746930/monotone-function-derivative-limit-bounded-differentiable-2?noredirect=1&lq=1 – co.sine Apr 21 '18 at 07:29

Monotone Function, Derivative Limit Bounded - Differentiable?

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