Is every differentiable function on $(0,1)$ uniformly continuous $?$

Question

$$f:(0,1)\rightarrow [0,1]$$ is a differentiable function . Is it uniformly continuous then $?$

Now $f$ being differentiable on $(0,1)$ is continuous , that is easy. Now I could say it is uniformly continuous if the differentiability on $(0,1)$ implied continuity on $[0,1]$. What the range being the closure of the domain contributes here $?$

Please give some lead on how to proceed .

Thanks for any help.

Answer 1

If we consider $$f(x)=\frac{1+\sin(1/x)}{2},$$ we get a differentiable function that is not uniformly continuous.