The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c(x) goes from 0 to 1 as c(x)=\mu([0,x]). This probability distribution, called the Cantor distribution, has no discrete part. That is, the corresponding measure is atomless. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.
