Proving a function is not uniformly continuous I have a homework question to which is 

If $f(x)$ is diferentiable in $(0,\infty)$  and $f'(x)>\frac{1}{x}$
  for every $x>0$ then $f$ is not uniformly continuous in $(0,\infty)$.

The question has a clue which asks me to prove that for $0<a<b$ then $f(b)-f(a)>\frac{b-a}{b}$ which I have also not managed to prove.
But even if I did I can't how to use that fact to solve the question. 
Can someone help me out? Thanks a lot :) 

Hints: 
1) Use the Mean Value Theorem to obtain the inequality.
2) Set $b=2a$. In this case $f(b)-f(a)>1/2$. So, can you choose $a$ and $b$ as closely as you like and still have $|f(b)-f(a)|$ big?
