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I can construct many function continuous in this interval . I have to choose one function that is not differentiable in $(0,1)$ because all point of $\{\frac{1}{1+n}: n \in\mathbb N\}$ is in the interval $(0,1)$,but i don't know that function choose that is not differentiable,moreover I don't know like prove that one function is not differentiable, with the definition or have to use other theorem?

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    $\begingroup$ take triangls with sommit at $1/(n+1) $.like mountains. $\endgroup$ May 25, 2017 at 23:12
  • $\begingroup$ Restrict the domain of your favorite everywhere-continuous, nowhere-differentiable function :P $\endgroup$
    – Kaj Hansen
    May 25, 2017 at 23:13

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I am not sure that I understood the question, but I suspect that $\left|\sin\left(\frac\pi x\right)\right|$ will do.

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You may apply condensation of singularities. The function $f(x)=\left\{\begin{array}{rcl} 0 & \text{at } x=0\\ x\sin\frac{1}{x} & \text{otherwise}\end{array}\right.$ is everywhere continuous and differentiable everywhere except at the origin. We have $\left|f(x)\right|\leq 1$, hence

$$ g(x)=\sum_{n\geq 1}\frac{1}{2^n}\,f\left(x-\frac{1}{n}\right) $$ is a continuous function on $[0,1]$, but it is not differentiable at any point of the form $\frac{1}{n}$.

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