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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)≤1x 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
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)≤1x 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