Pointwise convergence

A sequence of functions f_n converges pointwise if f_n(x)→f(x) for each x.

Pointwise convergence

Let $X$ be a set and let $(Y,d_Y)$ be a metric space . A sequence of functions $f_n:X\to Y$ converges pointwise to a function $f:X\to Y$ if

\forall x\in X,\quad \lim_{n\to\infty} d_Y\bigl(f_n(x),f(x)\bigr)=0.

One writes $f_n(x)\to f(x)$ for each fixed $x$ .

Pointwise convergence is the weakest common mode of convergence for functions. Many analytic properties (continuity , integrability, differentiation) are not preserved under pointwise limits without additional hypotheses. Compare with uniform convergence .

Examples:

On $[0,1]$ , let $f_n(x)=x^n$ . Then $f_n(x)\to f(x)$ pointwise, where $f(x)=0$ for $0\le x<1$ and $f(1)=1$ .
If $f_n(x)=\frac{1}{n}\sin x$ on $\mathbb{R}$ , then $f_n\to 0$ pointwise (and uniformly).
If $f_n=\mathbf{1}_{(0,1/n)}$ on $\mathbb{R}$ , then $f_n\to 0$ pointwise, but the “mass” near $0$ illustrates why pointwise convergence can miss uniform control.