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Uniform convergence guarantees pointwise convergence at every point.
Uniform convergence implies pointwise convergence: Let be a set and let (or ) be functions. If uniformly on , then pointwise on ; that is, for every we have .
The converse generally fails: pointwise convergence alone is typically too weak to preserve analytic properties such as continuity.
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.