Proving that $f(x)$ isn't uniformly continuous...

Question

I'd like to prove this fact which I've read in my textbook:

Given $f(x)$ continuous and differentiable on $[c, +\infty)$, if

$\displaystyle \lim_{x \to \infty} f'(x) = \pm \infty$

then $f(x)$ isn't uniformly continuous.

There is a hint that says to use Lagrange Theorem, but I can't figure out how to use it.

Answer 1

Suppose it is uniformly continuous, for every $c>0$, there exists $d>0$ such that $|x-y|<2d$ implies that $|f(x)-f(y)|<c$. There exists $M$ such that $x>M$ implies that ${{|f'(x)|}}>{c\over d}$. Take $x>M$, $|f(x)-f(x+d)|=|f'(c_x)||d|>c$.