Pointwise convergence

Definition

Define for $n\in \mathbb {N} ,f_{n}:S\subset \mathbb {R} \to \mathbb {R}$. The sequence of real valued functions $(f_{n})_{n=1}^{\infty }$ converges pointwise to a function $f:S\to \mathbb {R}$ if for every $\varepsilon >0$ there exists $N\in \mathbb {N}$ such that for $n>N$ and for each $x\in S$ we have $|f_{n}(x)-f(x)|<\varepsilon$.

Properties

If a sequence of functions $(f_{n})_{n=1}^{\infty }$ converges pointwise to a limit $f$, then $f$ can be found by

$\lim _{n\to \infty }f_{n}(x)=f(x).$

Continuity

Pointwise convergence need not preserve continuity, for example define for $x\in [0,1]$

$f_{n}(x)=x^{n},$

and observe that the limit for $x\in [0,1)$

${\begin{aligned}f(x)&=\lim _{n\to \infty }f_{n}(x)\\&=\lim _{n\to \infty }x^{n}\\&=0\end{aligned}}$

and for $x=1$ we have

${\begin{aligned}f(1)&=\lim _{n\to \infty }f_{n}(1)\\&=\lim _{n\to \infty }1^{n}\\&=1\end{aligned}}$

which means that $f$ may be written

$f(x)={\begin{cases}0,0\leq x<1\\1,x=1.\end{cases}}$

This function is trivially not continuous, despite all the functions in the sequence being perfectly continuous over the domain. For a stronger form of convergence that preserves continuity, see uniform convergence.