If $f$ is differentiable at a point $x=a$ can we say that there should be some open interval containing $a$ and f is continuous there?

Question

My lecturer in some his videos says that but I cannot believe it without proof.

My intuition says that this is true because differentiating is local property and we should do in some open interval, and in this interval since f is differentiable f must be continuous?

Unfortunately, intuition fails sometimes. There are counterexamples to that. Take a look at this: https://math.stackexchange.com/questions/2050665/can-a-function-be-differentiable-at-only-isolated-points — Bryan Castro, May 14 '20 at 00:06

score 2 · Accepted Answer · answered May 14 '20 at 00:06

2

If $f(x)=\begin{cases}x,&\mbox{when $x$ is rational}\\ -x,&\mbox{when $x$ is irrational}\end{cases}\quad$ and we set $g(x)=xf(x),$

then $g$ is differentiable only at $x=0.$

answered May 14 '20 at 00:06

Matematleta

29,139

If $f$ is differentiable at a point $x=a$ can we say that there should be some open interval containing $a$ and f is continuous there?

1 Answers1