Suppose a real-valued function f, whose domain is an interval, has a left-limit and a right-limit at every point in the domain. Are the following true?

1: f has at most a countable number of discontinuities.

2: f is continuous between discontinuities.

3. f is piecewise continuous, with potentially a countable infinity of pieces

Suppose a real-valued function f, whose domain is an interval, has the property that at every point in its domain it has a left-limit and a right-limit, and it equals at least one of these. Is it true that

1: f has at most a countable number of discontinuities. (Young's theorem would appear to say "yes".)

2. f can be called piecewise continuous. (Some say "piecewise continuity" requires a finite number of discontinuities.)