Question about differentiability/continuity,please help

Question

I was reading in my textbook that it says "a function $ f $ may have a derivative $ f' $ which exists at every point, but is discontinuous at some point."

Before this there is a theorem that says that if a point is differentiable then it's also continuous. I think I'm missing something; how can this be true?

but how can this be true, because if $ f' $ exists at a point doesn't that point have to be continuous? — user42, Feb 11 '15 at 18:47
You are confusing the continuity of $f$ at some point (necessary for $f'$ at this point to exist) with the continuity of $f'$ at the same point. — Did, Feb 11 '15 at 18:57

score 1 · Accepted Answer · answered Feb 11 '15 at 18:49

1

The theorem says: if $f'$ exists at a point, then $f$ is continuous at that point (but $f'$ may or may not be continuous).

The example says: there is a function $f$ that has a derivative $f'$ and $f'$ is not continuous (but $f$ is).

answered Feb 11 '15 at 18:49

Julián Aguirre

76,354

but if the $ f $ is continuous doesn't $ f' $ also have to be continuous? – user42 Feb 11 '15 at 18:50
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Is there an example of a differentiable continuous function with a discontinuous derivative? – user42 Feb 11 '15 at 18:51
@user1961722: video – user66081 Feb 11 '15 at 18:53
ok thanks everyone – user42 Feb 11 '15 at 19:04

score 0 · Answer 2 · edited Apr 13 '17 at 12:21

0

Have you already read the following answer? Probably not and I advice you to do so: discontinuous derivative.

edited Apr 13 '17 at 12:21

Community

1

answered Feb 11 '15 at 18:53

Ok thanks. I'm having a hard time intuitively understanding why this can be true but the example makes sense. – user42 Feb 11 '15 at 18:59

Question about differentiability/continuity,please help

2 Answers2