  We study a class of high frequency path functionals of a diffusion with a
singular threshold, including the case of a sticky-reflection, and establish
convergence to the local time. These functionals are built upon a test function
and a normalizing sequence. This advances existing results on sticky,
oscillating (regime-switching) and skew or reflecting diffusions in several
directions. First, it considers any normalizing sequence that diverges slower
than $n$. Second, it establishes the result for a sticky-oscillating-skew (SOS)
threshold. Based on this, and an approximation of the occupation time, we
devise consistent skew and stickiness parameter estimators.
