Q:

Does there exist a nowhere differentiable, everywhere continous, monotone somewhere function?

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. 

A:

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. 

