Real Analysis/Pointwise Convergence

←Generalized Integration

Real Analysis Pointwise Convergence

Uniform Convergence→

Let $f_{n}(x)\,$ be a sequence of functions defined on a common domain $D\subseteq \mathbb {R} \,$. Then we say that $f_{n}(x)\,$ converges pointwise to a function $f(x)\,$ if for each $x\in D\,$ the numerical sequence $f_{n}(x)\,$ converges to $f(x)\,$. More precisely speaking:

For any $x\in D\,$ and for any $\varepsilon >0\,$, there exists an N such that for any n>N, $\left|f_{n}(x)-f(x)\right|<\varepsilon$

An example:

The function

$f_{n}(x)={\frac {x^{n}}{1+x^{n}}}$ converges to the function

$f(x)=\left\{{\begin{array}{ll}1&{\text{if }}|x|>1\\{\frac {1}{2}}&{\text{if }}x=1\\0&{\text{if }}|x|<1\\\end{array}}\right.$

This shows that a sequence of continuous functions can pointwise converge to a discontinuous function.