real analysis - Antiderivative of discontinuous function

I am having confusion regarding anti-derivative of a function.

$$f(x) = \left\{\begin{array}{ll} -\frac{x^2}{2} + 4 & x \le 0 \\ \phantom{-} \frac{x^2}{2} + 2 & x > 0 \end{array} \right.$$

Consider the domain $[-1, 2]$.
Clearly the function is Riemann integrable as it is discontinuous at finite number of point. However is there a function $g(x)$ such that $g'(x) = f(x) \forall x \in [-1,2]$ ?

Answer

There is no such function, because by Darboux's theorem (cf. http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis) ), every derivative has to fulfill the intermediate value theorem, but $f$ does not.