# Darboux's theorem (analysis)

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

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

## Proof

$\phi (t)=f(t)-yt.$ Another proof based solely on the mean value theorem and the intermediate value theorem is due to Lars Olsen.

## Darboux function

A Darboux function is a real-valued function f which has the "intermediate value property": for any two values a and b in the domain of f, and any y between f(a) and f(b), there is some c between a and b with f(c) = y. By the intermediate value theorem, every continuous function is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions.

Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the left hand and right hand limits does not exist.

An example of a Darboux function that is discontinuous at one point, is the function $x\mapsto \sin(1/x)$ .

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function $x\mapsto x^{2}\sin(1/x)$ is a Darboux function that is not continuous.

An example of a Darboux function that is nowhere continuous is the Conway base 13 function.

Darboux functions are a quite general class of functions. It turns out that any real-valued function f on the real line can be written as the sum of two Darboux functions. This implies in particular that the class of Darboux functions is not closed under addition.

A strongly Darboux function is one for which the image of every (non-empty) open interval is the whole real line. Such functions exist and are Darboux but nowhere continuous.