Let function ƒ be Differentiable

[sergey_le]

Homework Statement

Let function ƒ be Differentiable in the interval [0,1] so that 0≤f'(x)≤1 for all x in the interval [0,1].
Prove that there is a point x in [0,1] so that f'(x)=x.

Relevant Equations

Intermediate value theorem

What I've tried is:
I have defined a function g(x)=f(x)-x^2/2. g Differentiable in the interval [0,1] As a difference of function in the interval.
so -x≤g'(x)≤1-x for all x∈[0,1] than -1≤g'(x)≤0 or 0≤g'(x)≤1 .
Then use the Intermediate value theorem .
The problem is I am not given that f' is continuous

 

[member 587159]

Interesting problem. First, it looks like you need continuity to apply the intermediate value theorem, but in fact you don't. Here you know that $f$ is differentiable, so $f'$ is still 'sufficiently nice', in the following sense:

$f'$ may be discontinuous, but it still satisfies the intermediate value theorem. This is known as Darboux's theorem. See https://en.wikipedia.org/wiki/Darboux's_theorem_(analysis) for two short elementary proofs.

 

[sergey_le]

Interesting problem. First, it looks like you need continuity to apply the intermediate value theorem, but in fact you don't. Here you know that $f$ is differentiable, so $f'$ is still 'sufficiently nice', in the following sense:

$f'$ may be discontinuous, but it still satisfies the intermediate value theorem. This is known as Darboux's theorem. See https://en.wikipedia.org/wiki/Darboux's_theorem_(analysis) for two short elementary proofs.

Thanks