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.