If there is a function that
continuous in a interval
monotonic in the same interval
Does it mean the function is also differentiation function in the interval?
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Conrado Costa
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Ohad Shamir
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5Look at $f(x)=\cases{ x, &$x<1$\cr 2x-1, &$x\ge 1$}$ on the interval $(0,2)$. – David Mitra Jul 09 '15 at 11:21
2 Answers
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By Lebesgue differentiation theorem, every monotonic function is almost everywhere differentiable.
However, we cannot replace "almost everywhere" with "everywhere": just consider $f(x)=x+\frac{1}{2}|x|$ in $x=0$.
Jack D'Aurizio
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No. There exists functions that are continuous on an interval, but differentiable nowhere. See details in Wikipedia: Weierstraß function.
If the function is also monotonic, a non-trivial theorem states that a monotonic function is differentiable, except perhaps on a subset of measure $0$. See Monotonic functions.
If you read French, you can see a proof that uses the Rising Sun Lemma here.
Bernard
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