The aim of this work is to extend the recent work of the author on the discrete frequency function to the more delicate continuous frequency function $\mathcal{T}$, and further to investigate its relations to the Hardy-Littlewood maximal function $\mathcal{M}$, and to the Lebesgue points. We surmount the intricate issue of measurability of $\mathcal{T}f$ by approaching it with a sequence of carefully constructed auxiliary functions for which measurability is easier to prove. After this, we give analogues of the recent results on the discrete frequency function. We then connect the points of discontinuity of $\mathcal{M}f$ for $f$ simple to the zeros of $\mathcal{T}f$, and to the non-Lebesgue points of $f$.