Lipschitz continuity

Imagine a function whose graph is never allowed to be steeper than some fixed slope — no matter where you look, it rises or falls at a bounded rate. That speed limit is exactly Lipschitz continuity: the change in output is controlled by a constant multiple of the change in input.

Formally, f is Lipschitz on a set D if there is a constant K >= 0, called a Lipschitz constant, with |f(x) - f(y)| <= K |x - y| for all x and y in D. The smallest such K measures the worst-case slope. If f is differentiable with |f'| <= K throughout an interval, then f is Lipschitz with that constant — by the mean value theorem.

Lipschitz continuity sits strictly between differentiability and uniform continuity: Lipschitz implies uniformly continuous (take delta = epsilon/K), but a Lipschitz function need not be differentiable (think of |x|, which is Lipschitz with K = 1 but has a corner at 0). And uniform continuity does not imply Lipschitz: sqrt(x) on [0, 1] is uniformly continuous yet its slope blows up at 0, so no finite K works. The condition is central to differential equations, where it guarantees unique solutions.