Continuity bans jumps: tiny input shifts, tiny outputs
A continuous function changes its value only a little when its input shifts a little—no sudden jumps allowed. Bernard Bolzano sketched an epsilon–delta style definition in 1817; Cauchy and later Weierstrass sharpened pointwise versus uniform continuity.
In calculus and analysis, continuity is core for real and complex functions and generalizes to maps between metric and topological spaces. Uniform continuity is a stronger global demand; order theory adds related notions such as Scott continuity. Everyday intuition compares a flower's height over time (continuous) with a bank balance that leaps when deposits clear (discontinuous).
Bolzano's 1817 contribution anticipated epsilon–delta control of closeness. Cauchy spoke of infinitely small increments producing infinitely small changes, paralleling infinitesimal readings still discussed today. Bolzano also separated pointwise from uniform continuity in the 1830s, though that manuscript waited until the 1930s to appear; Weierstrass likewise insisted on rigorous limits.
Pointwise continuity still has three competing, nonequivalent definitions in use. Weierstrass required the function's value and both one-sided limits to exist and agree; Édouard Goursat needed the value to match only one side; Camille Jordan allowed continuity even when the function was defined on just one side of the point. Eduard Heine published the first definition of uniform continuity in 1872, building on lectures Dirichlet gave in 1854. For real functions the picture is a graph drawn as one unbroken curve, and every polynomial is continuous everywhere. Functions like the reciprocal, undefined at a single point, may be called continuous or discontinuous depending on context. The gap in x times sine of 1 over x at zero is removable, since its limit there is 0, whereas sine of 1 over x has no limit at zero and cannot be patched.
Source: Continuous function
