About Darboux theorem What I have learned says that Darboux theorem can be used to prove function that is not continuous can also be true with intermediate value theorem. Is that right? But why don't we talk about that topic when we prove this theorem? Is this not that important? Please answer.....
p.s I sorry that I'm not good at english... and I'm highschool student in Korea. Can you explain more easier than others?
 A: Consider any differentiable function $f(x)$.
So, in the special case where $f(x)$ is continuously differentiable, then of course the derivative $f'(x)$ is. continuous, and hence, by intermediate value theorem(IVT), will attain all values between it's bounds. To reiterate, in this special case where our function is continuously differentiable, yes Darboux's theorem is simply the usual IVT.
Now, in the more general case,  where continuity of $f'(x)$ is not guaranteed, we cannot apply our usual IVT. But because $f(x)$ is differentiable, we can still apply Darboux's theorem, which does give us what IVT would've given if it were applicable. The difference of course is that the proof of IVT wouldn't have worked as it's hypothesis wasn't satisfied, but Darboux's proof works as it has different requirements than IVT.
Finally, to summarise, if you have a function, and know that it's a derivative of some function, you can have the result of IVT even if this function is itself not continuous, thanks to Darboux. Of course in case the function is continuous, you may directly use IVT not caring whether this function is a derivative of some function or not.
P.S. My apologies for redundancies, but I wanted to ensure that I get across the language gap.
