  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 $f$.
