Real analysis/Pointwise Convergence

Let f_n(x) be a sequence of functions. Then f_n(x) converges pointwise to a function f(x) if for any x, for any ε, there exists an N such that for any n>N, that |f_n(x)-f(x)|<ε.

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.