Continuity and uniform continuity

For continuity at x, choose the desired output tolerance epsilon and then choose a positive input tolerance delta. That delta may depend on x as well as epsilon. The claim concerns all domain points within that distance of x, including only points actually belonging to the domain. A function on a discrete metric space is continuous without being smooth or following a familiar algebraic formula: a sufficiently small ball contains just the chosen point. Continuity is about the domain and target metrics, rather than about how pleasant a graph looks.

Uniform continuity chooses delta before either input point is selected. Once epsilon is fixed, the same positive delta must control every pair in the domain. Every uniformly continuous function is continuous, but the converse fails on general domains. Another equivalent continuity criterion in metric spaces is preservation of convergent sequences: whenever x_n approaches x within the domain, f(x_n) approaches f(x). To disprove continuity, a single carefully chosen convergent sequence with bad image behavior is enough.

A Lipschitz bound compares output distance with input distance using one finite constant L. When L is positive, taking delta=epsilon/L proves uniform continuity. A constant function is handled separately because every input tolerance works. On a bounded interval [−M,M], the square function is Lipschitz with constant 2M, using the factorization of the difference of squares. This argument does not require differentiability and makes the role of the bounded domain explicit. When M is zero, the domain has one point and the conclusion is immediate.

On the whole real line, squaring is not uniformly continuous. Compare x_n=n with y_n=n+1/n. Their input difference tends to zero, while their squared output difference is 2+1/n². A hypothetical global tolerance for output error one would eventually accept these arbitrarily close inputs but fail its own output requirement. The construction is stronger than saying the derivative is unbounded: an unbounded derivative alone is not a general universal test without additional hypotheses, whereas these explicit pairs directly contradict the definition.

A continuous map on a compact metric domain is uniformly continuous. One useful proof by contradiction begins with arbitrarily close input pairs whose outputs stay a fixed positive distance apart. Compactness gives a convergent subsequence of one member of each pair. The pair distances tending to zero force the other members to approach the same limit. Continuity then sends both image subsequences to the same point, contradicting their persistent separation. Compactness prevents the problematic inputs from escaping without a limit in the domain.

Uniformly continuous maps send Cauchy sequences to Cauchy sequences. Given an output tolerance, obtain a common input tolerance and apply the Cauchy condition to the original tail. This is why uniform continuity supports extension to completions. Mere continuity cannot replace it: the sequence 1/n is Cauchy in (0,1], but its image under 1/x is n, which is not Cauchy. Uniform continuity need not imply a linear Lipschitz bound; the square-root function on [0,1] is uniformly continuous despite unbounded sensitivity near zero.