# Thread: Derivative Increasing ==> Derivative Continuous

1. ## Derivative Increasing ==> Derivative Continuous

$f$ is differentiable on $[a,b]$. Prove if $f'$ is increasing on $(a,b)$, then $f'$ is cont. on $(a,b)$.

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

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