Real Analysis/Pointwise 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.

Análise real/Índice/Convergência pontual