Monotonic function only has jump discontinuities I'm trying to show that a monotone function on a closed interval can only contain jump discontinuities. Could someone give me a hint as to how I should begin? I am not sure how to start this problem.
Edit: Let $f$ be a increasing function. Then if $x \leq y, f(x) \leq f(y)$.

Hint


*

*Reduce the problem to the case of proving for increasing functions by observing that if $f$ is decreasing, $-f$ is increasing. 

*Show that the one sided limits exists and are given by: $$\lim_{t \to c^+}f(t)=\inf\{f(x) \mid x \gt c\}\quad \mbox{and} \quad \lim_{t \to c^-}f(t)=\sup \{f(x) \mid x \lt c\}$$

*Since one-sided limits exist, the discontinuity is either removable or a jump discontinuity. Can you prove that it cannot be a removable discontinuity?
