Is there a nowhere differentiable but continuous everywhere function which is monotone in some small interval however small it is?
Until now I have seen only the Weierstrass function and it seems to be oscillating everywhere.
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3$\begingroup$ No. Monotone on an interval implies differentiable almost everywhere on that interval. $\endgroup$– ocgMar 20, 2014 at 16:28
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$\begingroup$ @JulienGodawatta can you please post a proof of the result $\endgroup$– happymathMar 20, 2014 at 16:31
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$\begingroup$ @DanielFischer thank you I have not yet studied differentiation in the lebesgue context so is there some intuitive way to see it $\endgroup$– happymathMar 20, 2014 at 16:38
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$\begingroup$ I expect there is a more elementary argument for monotonic functions, but I don't see it at the moment, sorry. $\endgroup$– Daniel FischerMar 20, 2014 at 16:47
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2$\begingroup$ For those interested, Rubel's 1963 paper Differentiability of monotonic functions is possibly the most elementary (in the sense of mathematical machinery used) proof I've seen. (No vitali covering theorem, for instance.) Also, Riesz/Nagy's book Functional Analysis may have a simple proof for the case when the monotone function is continuous, which is enough for the issue being discussed here, but my copy is at home and I'm elsewhere. $\endgroup$– Dave L. RenfroMar 20, 2014 at 17:13
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1 Answer
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It is well known that if $f : [a; b] \to R$ is increasing, then f is differentiable a.e., and $\int_a^bf^{'}(x)dx \le f(b)-f(a)$(see, for example,
http://www.math.ntnu.no/~quigg/4225/notes/differentiation.pdf, Theorem 1.6).
Hence, there does not exist such a function which is a nowhere differentiable but continuous everywhere and monotone in some small interval.
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$\begingroup$ The link is broke. Would you mind mending it? Thank you. $\endgroup$– HansApr 25, 2021 at 3:03