Property of a function at a point (or on a set) that arbitrarily small changes in the input produce arbitrarily small changes in the output; formally, in a metric context: for each point x0 and each ε>0 there exists δ>0 such that d(x,x0)<δ implies d(f(x),f(x0))<ε.

Principle

Local control of outputs by inputs: near each point the function's variation can be bounded by choosing a sufficiently small neighborhood determined by the point and the desired output tolerance.

Demonstration

The sine function on R is continuous everywhere: for any x0 and ε>0 one can find δ depending on x0 and ε so that |sin x - sin x0|<ε whenever |x-x0|<δ. The Heaviside step function is discontinuous at 0.

Misapplication

Treating continuity as if it guaranteed uniform smallness of variation over the whole domain (confusing pointwise continuity with uniform continuity), or assuming continuity implies boundedness on noncompact domains.

Consequence

Continuous functions preserve limits and connectedness, and on compact domains they are bounded and attain maxima and minima. Continuity is preserved under sums, products, and composition.

Reversal

A discontinuous function has at least one point where arbitrarily small input changes can produce a jump or other nonvanishing change in output; discontinuity is the negation of the local ε–δ property.

Boundary

Requires a topology or metric to compare 'small' changes; statements about continuity must specify domain and codomain (metric, topological or uniform structure). Continuity as used here excludes weaker, measure-theoretic notions like almost-everywhere continuity unless stated.

Semantic Tension

Often confused with uniform continuity and with differentiability: continuity is strictly weaker than either, and the same word is used for both pointwise and global variants, which creates ambiguity without further qualification.

Synthesis

Continuity is the local ε–δ condition guaranteeing that outputs vary arbitrarily little when inputs are restricted to a small neighbourhood of a point; it is the basic continuity of behaviour on which stronger global or smoothness properties build.