Derivative Increasing ==> Derivative Continuous

**ABigSmile** · February 22nd 2010, 07:18 AM

$f$ 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.

**Drexel28** · February 23rd 2010, 07:54 AM

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.

**Jose27** · February 23rd 2010, 10:58 AM

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?

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