Let $f'$ be continuous. If $f$ is uniformly continuous then $f'$ is uniformly continuous? Is this, in general, true? I need this in order to prove that a polynomial is uniformly continuous in $\mathbb{R}$ if and only if $p(x)$ is of degree less or equal to 1. If $p(x)$ is of degree less or equal to 1, is easy to prove that is uniform. Now, if $p(x)$ is of degree greater than one, then is not uniform. I want to proceed by induction, for degree greater or equal to 2, if $p(x)$ is of degree 2, is not uniforn. Then by induction $p'(x)$ is not uniform, and therefore $p(x)$ is not uniform either. 

This is a modification of the example by Guido A wherein the function is defined on the whole line: take $f(x)=\sqrt {\frac {\pi} 2 +\tan^{-1}x}$. [Note that $f'$ is unbounded].
