# Real Analysis/Pointwise Convergence

 Real Analysis Pointwise Convergence

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

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


An example:

The function

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

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

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