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!
 A: If I understood the OP correctly, he wants some simple examples of functions, which are not continuous and they have Darboux property. (He wants to practice showing that a function has intermediate value property on some concrete examples.) 
I've given a few examples. I have made this post CW, so feel free to add further examples.

Functions which are not continuous, but are derivatives:
$f(x)=
  \begin{cases}
    \sin\frac1x, & x\ne 0, \\
    0 & \text{otherwise}.
  \end{cases}
$
$g(x)=
  \begin{cases}
    2x\cos\frac1x+\sin\frac1x, & x\ne 0, \\
    0 & \text{otherwise}.
  \end{cases}
$
$h(x)=
  \begin{cases}
    2x\sin\frac1{x^2}-2\frac1x\cos\frac1{x^2}, & x\ne 0, \\
    0 & \text{otherwise}.
  \end{cases}
$
The functions $f(x)$, $g(x)$ are $h(x)$ are from the book Van Rooij-Schikhof: A Second Course in Real Analysis (in the Introduction.).

Functions which are not continuous, but have Darboux property (intermediate value property):
$f_2(x)=
  \begin{cases}
    \sin\frac1x, & x\ne 0, \\
    1 & \text{otherwise}.
  \end{cases}
$
Again from the book Van Rooij-Schikhof: A Second Course in Real Analysis (in the Introduction.).
