# Thread: Derivative Increasing ==> Derivative Continuous

1. ## 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.

2. 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 $\displaystyle x \in (a,b)$.
$\displaystyle f$ 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.