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A uniform limit of continuous functions between metric spaces is continuous.
Let and be metric spaces, and let be continuous for every . If uniformly on , then is continuous.
This preservation property distinguishes uniform convergence from pointwise convergence, which alone does not preserve continuity.
Let be a function between metric spaces, and let . We say is continuous on if the restriction , with carrying the subspace metric, is continuous. Equivalently, is continuous at every point .
Spelled out: for every and every , there exists such that for all ,
Let be a set and let be a metric space. A sequence of functions converges uniformly to if
Equivalently,
where the supremum is taken in .
Uniform convergence is strong enough to pass many properties to the limit (e.g., continuity, under standard hypotheses). Compare with pointwise convergence. It is a central tool in analysis and approximation theory.
A sequence of functions from a set into a metric space converges uniformly to a function if for every there exists such that for all and all ,
Equivalently,
Pointwise convergence of a sequence of functions to a function on a set means: for every , the real sequence converges to , i.e.
This is a basic mode of convergence for sequences of functions, and it is weaker than uniform convergence (where one works simultaneously for all ). For each fixed , pointwise convergence is just ordinary convergence of a sequence in the codomain.