# Derivative Increasing ==> Derivative Continuous

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• Feb 22nd 2010, 07:18 AM
ABigSmile
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.
• Feb 23rd 2010, 07:54 AM
Drexel28
Quote:

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

Read that again.
• Feb 23rd 2010, 10:58 AM
Jose27
Quote:

Originally Posted by ABigSmile
$f$ 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.

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