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?

  • 2
    $\begingroup$ 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

2 Answers 2


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


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 $\frac{1}{n}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.