one-sided limit

Sometimes a function behaves like a staircase or a coastline: approach the point from the left and you head to one place, approach from the right and you head somewhere else. A one-sided limit looks at only one direction of approach, ignoring whatever happens on the other side.

The right-hand limit, lim_{x->a+} f(x) = L, requires |f(x) - L| < epsilon whenever 0 < x - a < delta — that is, x approaches a only from above. The left-hand limit, lim_{x->a-} f(x), is the mirror image, with -delta < x - a < 0. Each is an ordinary epsilon-delta limit restricted to one side.

The two-sided limit exists if and only if both one-sided limits exist and are equal; their common value is then the limit. When the one-sided limits exist but disagree, the function has a jump discontinuity at a. One-sided limits are also the natural notion at an endpoint of an interval, where approach is only possible from inside.