# Derivative Increasing ==> Derivative Continuous

• Feb 22nd 2010, 07:18 AM
ABigSmile
Derivative Increasing ==> Derivative Continuous
$\displaystyle f$ is differentiable on $\displaystyle [a,b]$. Prove if $\displaystyle f'$ is increasing on $\displaystyle (a,b)$, then $\displaystyle f'$ is cont. on $\displaystyle (a,b)$.

Ok since $\displaystyle f$ is diff. on $\displaystyle [a,b]$ it's also continuous on $\displaystyle [a,b]$. Since it's increasing we have $\displaystyle f'(x) \ge 0$ for any $\displaystyle x \in (a,b)$. The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that $\displaystyle f'$ is increasing to show it's derivative is continous? I struggle implementing MVT into my proofs apparently.
• Feb 23rd 2010, 07:54 AM
Drexel28
Quote:

Originally Posted by ABigSmile
Prove if $\displaystyle f'$ is increasing on $\displaystyle (a,b)$, . Since it's increasing we have $\displaystyle f'(x) \ge 0$ for any $\displaystyle x \in (a,b)$.

• Feb 23rd 2010, 10:58 AM
Jose27
Quote:

Originally Posted by ABigSmile
$\displaystyle f$ is differentiable on $\displaystyle [a,b]$. Prove if $\displaystyle f'$ is increasing on $\displaystyle (a,b)$, then $\displaystyle f'$ is cont. on $\displaystyle (a,b)$.

Ok since $\displaystyle f$ is diff. on $\displaystyle [a,b]$ it's also continuous on $\displaystyle [a,b]$. Since it's increasing we have $\displaystyle f'(x) \ge 0$ for any $\displaystyle x \in (a,b)$. The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that $\displaystyle f'$ is increasing to show it's derivative is continous? I struggle implementing MVT into my proofs apparently.

If you know the IVT for derivatives then just answer this: What kind of discontinuities can a monotone function have?