Is there a term for a periodic function that has non-discontinuous end behavior?

Question

Basically, a periodic function where the period boundary is $p$

$\lim_{x \to p^-} \left[ f(x) \right]= \lim_{x \to p^+} \left[ f(x) \right]$

As an example, a sine wave would qualify, but not a sawtooth wave.

Answer 1

I suspect the term you're looking for does not exist (or at least is not used) because there isn't really any particular "period boundary" of a periodic function. The period can start and stop anywhere. For example, $[0,2\pi]$ is a period of the sine function, but so is $[-\pi,\pi].$

Also, just as a sinusoidal function can be shifted left or right and is still a sinusoidal function, a sawtooth function can be shifted left or right and is still a sawtooth (and fits your description in every translated position except one).

But if you say you have a periodic function $f(x)$ that is continuous at $x=0,$ then since the function is periodic it is also continuous at $x=p,$ at $x=2p,$ at $x = -p,$ etc., where $p$ is the period of the function. "Periodic function continuous at zero" is a reasonably succinct description and would be correctly understood by anyone who knows periodic functions.