Real Analysis/Pointwise Convergence

Let $f_n(x)$ be a sequence of functions defined on a common domain $D\subseteq ℝ$. 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 $\epsilon >0$, there exists an N such that for any n>N, $|f_n(x)-f(x)|<\epsilon$

An example:

The function

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

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

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