Compactness in the Space of Quasi-Continuous Functions

It is well known that if C is the class of continuous functions on [a, b] with l.u.b. norm, then a subset F of C is compact in the Weierstrass-Bolzano sense (every infinite subset contains a sequence converging uniformly) if and only if the functions f of F are uniformly bounded and equicontinuous at all points of [a, b]. Only a slight change is needed to obtain a compactness condition in the space of quasi-continuous functions, those for which f(x +1O) and f (x-O) exist for all x of [a, b]. We have: