I am curious about a statement made on the wikipedia page about discontinuities. The article is http://en.wikipedia.org/wiki/Discontinuity_(mathematics) and the question is about the caption on the first picture to the right (I have posted the picture in question at the end of the post). The caption states, "The derivative of this curve has a jump discontinuity."
We have that Darboux's Theorem states, "If $f$ is differentiable on an interval $[a,b]$, and if $\alpha$ satisfies $f'(a)<\alpha <f'(b)$ (or $f'(a)>\alpha >f'(b))$, then there exists a point $c \in (a,b)$ where $f'(c)=\alpha$.
From Darboux's Theorem we can see that basically it means that any function with a jump discontinuity cannot be a derivative. If this is the case how can the derivative of the curve in the picture have a jump discontinuity?
Thank you for your help on clarification.