Irregular 1-sets on the graphs of continuous functions

One can define in a natural way irregular 1-sets on the graphs of several fractal functions, like Takagi’s function, Weierstrass-Cellerier type functions and the typical continuous function. These irregular 1-sets can be useful during the investigation of level-sets and occupation measures of these functions. For example, we see that for Takagi’s function and for certain Weierstrass-Cellerier functions the occupation measure is singular with respect to the Lebesgue measure and for almost every level the level set is finite.