Pathological functions that are continuous but differentiable only on a set of points of measure zero are sometimes known as monsters of real analysis. Examples include the Weierstrass and Blancmange functions.
Monsters of Real Analysis
See also
Blancmange Function, Continuous Function, Differentiable, Monster-Barring, Nowhere Differentiable Function, PathologicalExplore with Wolfram|Alpha
References
Chenevert, G. "Plotting the Monsters of
Real Analysis." http://library.wolfram.com/infocenter/MathSource/520/.Referenced on Wolfram|Alpha
Monsters of Real AnalysisCite this as:
Weisstein, Eric W. "Monsters of Real Analysis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MonstersofRealAnalysis.html