This is a question in Bergman's companion to Rudin's POA.
$f$ is differentiable on $[a,b]$ let g=f' Show that for $x \in (a,b], \ g(x-)\neq \infty \ \text{or} -\infty$
My suspicion is that if the derivative blows up to infinity it must drive the function itself to infinity which will destroy the continuity of the function but I'm having trouble proving this.
Alternative notation:
$f$ is differentiable on $[a,b]$
Show \lim_{t\rightarrow x-}f'(t) \neq \infty \ \text{or} -\infty for any $x \in (a,b]$