Differenttiable Function Proof

[Murk]

Suppose \(\displaystyle f\) is diff. on \(\displaystyle [a,b]\). Prove if \(\displaystyle f'\) is increasin on \(\displaystyle (a,b)\), then \(\displaystyle f'\) is continuous on \(\displaystyle (a,b)\). How should I approach this problem? I understand \(\displaystyle f\) is cont. on \(\displaystyle [a,b]\) and \(\displaystyle |f'(x)| > 0\). Should I start off with assuming \(\displaystyle f'\) is discontinuous on \(\displaystyle [a,b]\)? Or should I start off with the \(\displaystyle \epsilon - \delta\) approach? Any hints just to even get started correctly will help a ton.

 

[daon]

Suppose \(\displaystyle f\) is diff. on \(\displaystyle [a,b]\). Prove if \(\displaystyle f'\) is increasin on \(\displaystyle (a,b)\), then \(\displaystyle f'\) is continuous on \(\displaystyle (a,b)\). How should I approach this problem? I understand \(\displaystyle f\) is cont. on \(\displaystyle [a,b]\) and \(\displaystyle |f'(x)| > 0\). Should I start off with assuming \(\displaystyle f'\) is discontinuous on \(\displaystyle [a,b]\)? Or should I start off with the \(\displaystyle \epsilon - \delta\) approach? Any hints just to even get started correctly will help a ton.

Assume it is not continuous at some point x=c in (a,b)

let L = lim x-> c of f'(x) from the left
let R = lim x-> c of f'(x) from the right

there are a few cases

a)L=f'(c)=/=R,
b)L=/=f'(c)=R
c)L=/=f'(c)=/=R

with a and b you contradict that f is differentiable (show that there is a jump discontinuity in f'), and/or f' always increasing
with c you contradict that it is always increasing if L=R (think of a hole in the graph with f'(c) being either higher or lower than the limit... both cases should be considered).

otherwise if L is not equal to R there is an obvious jump discontinuity. as for jump discontinuities in f', think about how you'd prove |x| is not differentiable at 0.