In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
[[File:Drini-nonuniformconvergence.png|thumb|300px|
The pointwise limit of continuous functions does not have to be continuous: the continuous functions
<math>\sin^n(x)</math> (marked in green) converge pointwise to a discontinuous function (marked in red).]]
Suppose that <math>X</math> is a set and <math>Y</math> is a topological space, such as the real or complex numbers or a metric space, for example. A sequence of functions <math>\left(f_n\right)</math> all having the same domain <math>X</math> and codomain <math>Y</math> is said to converge pointwise to a given function <math>f : X \to Y</math> often written as
<math display=block>\lim_{n\to\infty} f_n = f\ \mbox{pointwise}</math>
if (and only if) the limit of the sequence <math>f_n(x)</math> evaluated at each point <math>x</math> in the domain of <math>f</math> is equal to <math>f(x)</math>, written as
<math display=block>\forall x \in X, \lim_{n\to\infty} f_n(x) = f(x).</math>
The function <math>f</math> is said to be the pointwise limit function of the <math>\left(f_n\right).</math>
The definition easily generalizes from sequences to nets <math>f_\bull = \left(f_a\right)_{a \in A}</math>. We say <math>f_\bull</math> converges pointwise to <math>f</math>, written as
<math display=block>\lim_{a\in A} f_a = f\ \mbox{pointwise}</math>
if (and only if) <math>f(x)</math> is the unique accumulation point of the net <math>f_\bull(x)</math> evaluated at each point <math>x</math> in the domain of <math>f</math>, written as
<math display=block>\forall x \in X, \lim_{a\in A} f_a(x) = f(x).</math>
Sometimes, authors use the term bounded pointwise convergence when there is a constant <math>C</math> such that <math>\forall n,x,\;|f_n(x)|<C</math> .
Properties
This concept is often contrasted with uniform convergence. To say that
<math display=block>\lim_{n\to\infty} f_n = f\ \mbox{uniformly}</math>
means that
<math display=block>\lim_{n\to\infty}\,\sup\{\,\left|f_n(x)-f(x)\right| : x \in A \,\}=0,</math>
where <math>A</math> is the common domain of <math>f</math> and <math>f_n</math>, and <math>\sup</math> stands for the supremum.