As it is well known every continuous function has the intermediate value property, but even some discontinuous functions like 
$$f(x)=\left\{
\begin{array}{cl}
\sin\left(\frac{1}{x}\right) & x\neq 0 \\
0 & x=0
\end{array}
\right.$$
are having this property.  

 In fact we know that a derivative always have this property. 

My question is, if for every function $f$ with the intermediate value property exists a function $F$ such that $F'=f$. And if so, is $F$ unique (up to a constant you may add)?

My attempts till now: (Just skip them if you feel so)

The most natural way of finding those functions would be integrating, so I guess the question can be reduced to, if functions with the intermediate value property can be integrated.  

This one depends heavy on how you define when a functions is integrable, we (my analysis class) said that we call a function integrable when it is a regulated function (the limits $x\to x\_0^+ f(x)$ and $x\to x\_0^- f(x)$ exists ) .  

 As my example above shows, not every function with the intermediate value property is a regulated function. But if we say a function is integrabel when every riemann sum converges the above function is integrable, so it seems like this would be a better definition for my problem. 

Edit: As Kevin Carlson points out in a commentar that being a derivative is different from being riemann integrable, he even gave an example for a function which is differentiable but it's derivative is not riemann integrable. So we can't show that those functions are riemann integrable as they are not riemann integrable in general. Now I have no clue how to find an answer.