# Does $f$ being continuous, differentiable imply $f'$ is continuous

Let $$f:[a,b] \rightarrow \mathbb R$$, continuous on $$[a,b]$$, differentiable on $$(a,b)$$. Does this imply $$f'$$ is continuous on $$(a,b)$$? I feel like this is true intuitively

• This is one of those classic questions that every good calculus student seems to have at some point in their lives <3 Oct 17 '19 at 4:20
• lol i guess i'm a "good calculus student". hooray Oct 17 '19 at 4:21
• If this question occurs to you while you're starting off studying calculus, I'd bet on you having a decent mind for learning math in the long run :) Oct 17 '19 at 4:35
• well I'm trying my best lol. I will say that as a stat major I have to take real analysis 1,2 and it has given me a whole new level of respect for math majors Oct 17 '19 at 6:34

Unfortunately, no. However $$f'$$ is severely constrained in this instance. It can be shown that $$f'$$ actually satisfies the conclusion of the intermediate value theorem. To give an example of this phenomenon, take $$f(x) = x^2 \sin(\frac{1}{x})$$ when $$x\ne 0$$, and $$f(0) = 0$$. Then $$f$$ is differentiable everywhere, but its derivative is not continuous at the origin.