Pointwise convergence

In analysis is the pointwise convergence property of a sequence of functions that converges for each element of the domain to the function value of a limit function, ie

We write then

Or

Formally converges to pointwise if

That is, there must be for each and for each natural number, such that for all.

Example

For example, the sequence converges with

In the interval pointwise to the function

Because obviously applies

Demarcation

It should however be noted that pointwise convergence is not equivalent to uniform convergence, because for example the last example above, namely pointwise, but not uniformly converges ( as is every member of the sequence continuously differentiable everywhere, the limit function, however, not even continuous): Uniform convergence is a much stronger statement.

For pointwise convergence, the values ​​of the functions fn are not necessarily real numbers, they may be elements of some topological space.