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I am curious about a statement made on the wikipedia page about discontinuities. The article is 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.

wikipedia picture in question

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The issue is that Darboux's theorem requires that $f$ be differentiable at every point in the interval. What's happening here is that the height function will be differentiable at every point except the x-value of the corner, so Darboux's theorem doesn't apply.

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So basically it is a statement about a piecewise function like that of |x| where we have to consider each section of the curve separately and then piecing back the derivatives gives us a jump discontinuity? – Differintegral Feb 27 '12 at 15:16
I think so - the derivative of the height function restricted to either the left or the right of the corner point will satisfy Darboux's theorem. – Jason DeVito Feb 27 '12 at 15:23

Strictly speaking, what you get is not a jump discontinuity in the derivative, but a point where the derivative is not defined at all -- and according to the article itself, the discontinuities it speaks of are when a function fails to be continuous "at a point in its domain".

That is not the case for the derivative at the point where it fails to exist. The image and its caption are therefore misplaced in this context.

You do get an actual jump discontinuity for each of the one-sided derivatives, but Darboux's theorem does not apply to one-sided derivatives.

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