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

$$f:(0,1)\rightarrow [0,1]$$ 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 $?$

• What about $\sin\frac1x$? – 5xum Sep 4 '15 at 18:43
• It is continuous , differentiable on $(0,1)$ but not uniformly continuous . Right $?$ – user118494 Sep 4 '15 at 18:47
• The derivative should be bounded, i.e. $|f'( \zeta)| \leq M$ for all $\zeta \in (0,1)$. Then, according to the mean value theorem, $|f(x) - f(x)| = |f(\zeta)| |x-y| \leq M |x-y|$. Which yields uniform continuity. I am not sure if this is the weakest possible assumption. – Nigel Overmars Sep 4 '15 at 18:53
• @NigelOvermars If you consider $f(x)=\sqrt x$ you get a uniformly continuous function with an unbounded derivative. – Kitegi Sep 4 '15 at 19:02
If we consider $$f(x)=\frac{1+\sin(1/x)}{2},$$ we get a differentiable function that is not uniformly continuous.