real analysis - Intermediate Value Property and Discontinuous Functions

This is a general question to which I need help finding a concrete example so that I may understand the concept/strategy better, and any help will be greatly appreciated.

If given a function $F$ that is not continuous, how can I show that the given function satisfies the intermediate value property? A hint that was given by the Professor was to find an auxiliary function $f$ such that $f'=F$.

I know that all continuous functions have the intermediate value property (Darboux's property), and from reading around I know that all derivatives have the Darboux property, even the derivatives that are not continuous.

Here is what I could make sense of the Professor's hint:

If I could find a suitable function $f$ which was differentiable, and $f'=F$, the derivative would have the intermediate value theorem (since all derivatives have the intermediate value property), and thus the original discontinuous function $F$ would also have the intermediate value theorem.

Can anyone please tell me if my reasoning is correct and/or please provide me with a discontinuous function that I can practice on (or perhaps direct me to another question with such a function that I may have overlooked)?

Thanks a lot in advance!