jump discontinuity
Think of a staircase, or a price that ticks up the instant the clock strikes noon. Approach the moment from before and the value heads one way; approach from after and it heads to a different value. The function leaps across a gap — that gap is a jump discontinuity.
Formally, f has a jump discontinuity at a if both one-sided limits exist and are finite but are unequal: lim_{x->a-} f(x) = L1 and lim_{x->a+} f(x) = L2 with L1 not equal to L2. The size of the jump is |L2 - L1|. Whatever value f(a) takes, no choice can repair the break, since the left and right approaches disagree.
A jump is a first-kind discontinuity, and unlike a removable one it cannot be fixed by redefining a single value. A clean fact worth remembering: a monotone function can only have jump discontinuities (its one-sided limits always exist), and it can have at most countably many of them — so its set of discontinuities, while possibly infinite, is never as large as an interval.
The Heaviside step H(x) = 0 for x < 0 and H(x) = 1 for x >= 0 has a jump discontinuity at 0: lim_{x->0-} H = 0 while lim_{x->0+} H = 1, a jump of size 1.
Unequal finite one-sided limits: a jump of size 1.