Nowhere diffferentable continuous function

[mathman]

TL;DR

Relationship between diffferentablity and monotonicity.

Weierstrass function is the classic example of a continuous function which is nowhere differentiable. What happens when a function is monotone? My guess that it cannot be nowhere differentiable. It seems to me the reverse is true - it is differentiable almost everywhere. Any light on the subject?

 

[Infrared]

Yes, this is true. Here's a proof: http://mathonline.wikidot.com/lebesgue-s-theorem-for-the-differentiability-of-monotone-fun

 

[mathman]

Thank you Infrared.