Do derivatives always exist in a neighborhood?

[slamminsammya]

Suppose that f:\mathbb{R} \to \mathbb{R} is continuous and f'(x_0) exists for some x_0 . Does it follow that f' exists for all x such that |x-x_0|<r for some r>0 ?

 

[JG89]

I asked the same question as you before:

https://www.physicsforums.com/showthread.php?t=306278

The answer is no.