If a function $y$ and its derivative $y'$ are defined on interval $I=[a,b]$,then does both $y$ and $y'$ are continuous on $I$?

Question

I was reading a lecture slide and there it was written that since $y$ and $y'$ are defined on a closed interval I,they must be continuous on $I$. I tried to know the reason behind it. I thought that since by Darboux theorem $y'$ will have intermediate value property in every possible closed sub interval of $I$. It may be due to that property that $y'$ is continuous. Is it so?

Answer 1

You shouldn't believe any more slides from the same source. The author is unaware of a very standard example: Define $$f(x)=\begin{cases}x^2\sin(1/x^5),&(x\ne0), \\0,&(x=0).\end{cases}$$

Then $f$ and $f'$ are defined everywhere, but $f'$ is not continuous at the origin.