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.

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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Let f: [0, 1] → [0, 1] be a continuous strictly increasing function.
(a) If f is absolutely continuous on [0, 1], prove that for any open set UC [0, 1] we have
\ƒ(U)| = √√ ƒ'(x) dx .
So
U
(b) Give an example that (1) is not true without the assumption of absolute continuity.
Transcribed Image Text:Let f: [0, 1] → [0, 1] be a continuous strictly increasing function. (a) If f is absolutely continuous on [0, 1], prove that for any open set UC [0, 1] we have \ƒ(U)| = √√ ƒ'(x) dx . So U (b) Give an example that (1) is not true without the assumption of absolute continuity.
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