Definitions

# Darboux's theorem (analysis)

Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions which result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval.

Note that when f is continuously differentiable (f in C1([a,b])), this is trivially true by the intermediate value theorem. But even when f′ is not continuous, Darboux's theorem places a severe restriction on what it can be.

## Darboux's theorem

Let f : [a,b] → R be a real-valued continuous function on [a,b], which is differentiable on (a,b), differentiable from the right at a, and differentiable from the left at b. Then $f,\text{'}$ satisfies the intermediate value property: for every t between $f,\text{'}_\left\{+\right\}\left(a\right)$ and $f,\text{'}_\left\{-\right\}\left(b\right)$, there is some x in [a,b] such that $f,\text{'}\left(x\right) = t$.

## Proof

Without loss of generality we might and shall assume $f,\text{'}_\left\{+\right\}\left(a\right) > t > f,\text{'}_\left\{-\right\}\left(b\right)$. Let g(x) := f(x) - tx. Then $g\text{'}\left(x\right) = f,\text{'}\left(x\right) - t$, $g\text{'}_\left\{+\right\}\left(a\right) > 0 > g\text{'}_\left\{-\right\}\left(b\right)$, and we wish to find a zero of $g\text{'}$.

Since g is a continuous function on [a,b], by the extreme value theorem it attains a maximum on [a,b]. This maximum cannot be at a, since $g\text{'}_\left\{+\right\}\left(a\right) > 0$ so g is locally increasing at a. Similarly, $g\text{'}_\left\{-\right\}\left(b\right) < 0$, so g is locally decreasing at b and cannot have a maximum at b. So the maximum is attained at some c in (a,b). But then $g\text{'}\left(c\right) = 0$ by Fermat's theorem (stationary points).