Showing that a functions derivative is not bounded on $\mathbb{R}$ Suppose that $f$ is differentiable but not uniformly continuous on $\mathbb{R}$. Prove that $|f'|$ is not bounded on  $\mathbb{R}$. 
So I know that to show that $|f'|$ isn't bounded you would have to show that for any constant B, $|f'(a)|$ $>$ $B$, for some $a$ $\in$ $\mathbb{R}$. How would I go about doing this?

Hint: Use the counterpositive. 
Claim: If $f: I \to \mathbb R$ is differentiable such that $|f'(x)| < M$, $\forall x \in I$, for some $M > 0$. Then $f$ is uniformly continuous. 
Proof: Use the Mean Value Theorem and show that it is Lipschitz.
