Let f: [0, 1] → [0, 1] be a continuous strictly increasing function. (a) If ƒ is absolutely continuous on [0, 1], prove that for any open set UC [0, 1] we have \ƒ(U)] = √√ f'(x) dx . (b) Give an example that (1) is not true without the assumption of absolute continuity.

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A: Given that: f is differentiable at xo   To prove: limh→0f(x0+h)-f(x0-h)2h=f'(x0)

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A: This question is based on Rolle's theorem which is explained in step 2.

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A: Hi, you can find the answer in the handwritten form.  Given f(a)=-2  f(b)= 2

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A: As per our guidelines we can answer only 3 subparts. please ask others seperately.

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