Editing Pointwise convergence (section)

==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 [[Infimum and supremum|supremum]]. That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.  For example, if <math>f_n : [0,1) \to [0,1)</math> is a sequence of functions defined by <math>f_n(x) = x^n,</math> then <math>\lim_{n\to\infty} f_n(x) = 0</math> pointwise on the interval <math>[0, 1),</math> but not uniformly.

The pointwise limit of a sequence of [[continuous function]]s may be a discontinuous function, but only if the convergence is not uniform. For example,
<math display=block>f(x) = \lim_{n\to\infty} \cos(\pi x)^{2n}</math>
takes the value <math>1</math> when <math>x</math> is an integer and <math>0</math> when <math>x</math> is not an integer, and so is discontinuous at every integer.

The values of the functions <math>f_n</math> need not be real numbers, but may be in any [[topological space]], in order that the concept of pointwise convergence make sense.  Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in [[metric space]]s, and, more generally, in [[uniform space]]s.