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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Sign up to join this communityIs 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.
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.