Monotone functions have only countably many discontinuities, and every one is a jump.Suggest a slogan
Let \(a<b\) and let \(f\colon(a,b)\to\RR\) be monotone. Then the set \(E\subseteq(a,b)\) of points at which \(f\) is discontinuous is at most countable. Moreover, every \(x\in E\) is a jump discontinuity: the one-sided limits \[ f(x-)\coloneqq\lim_{t\to x^-}f(t)\qquad\text{and}\qquad f(x+)\coloneqq\lim_{t\to x^+}f(t) \] both exist. [@Rud76].