Pointwise convergence

A sequence of functions converges pointwise on a domain when, at each fixed point, its values converge to the corresponding value of a limit function. The convergence threshold may depend on the point.

Sequence of functions: An ordered list of functions indexed by the natural numbers or another index set. Pointwise convergence describes the limiting behavior of such a sequence.
Uniform convergence: Convergence of functions whose index threshold can be chosen independently of the domain point. Unlike pointwise convergence, it controls approximation uniformly over the entire domain.
Pointwise limit: The function whose value at each point is the limit of a sequence of functions there. It is the function produced by pointwise convergence.
Fourier series: A representation of a periodic function as a sum of sine and cosine terms. Questions about convergence at each point are central to interpreting Fourier series.
Limit of a sequence: A value that the terms of a sequence approach as the index grows. At each domain point, function values must have this ordinary sequence limit.
Almost-everywhere convergence: Convergence at every point except possibly on a set of measure zero. It permits exceptional points, while pointwise convergence requires convergence at every point.
Continuous function: A function whose values change continuously with changes in its input. Pointwise limits of continuous functions need not remain continuous.
Weierstrass approximation theorem: The theorem that every continuous function on a closed bounded interval is uniformly approximable by polynomials. Its stronger uniform conclusion guarantees pointwise convergence of polynomial approximations.
Epsilon-delta definition of a limit: A formal definition of a limit using quantified bounds on distances between inputs and outputs. Its epsilon-N form states exactly what convergence at each fixed point requires.
Convergence in measure: Convergence in which the measure of the set with large error tends to zero. It measures the size of error sets rather than requiring pointwise limits.

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