Pointwise Convergence · aiwiki.page

Pointwise convergence is a form of convergence for sequences of functions in which the function values approach a limit separately at every fixed point of a common domain. In mathematical analysis, it provides a basic way to define a limiting function from successive approximations. Its defining feature is that the index beyond which an approximation achieves a prescribed accuracy may depend on the point being evaluated. This distinguishes it from uniform convergence, which requires a single index to work throughout the domain. (jirka.org)

Definition

Let $f_n:X\to Y$ be functions with common domain $X$ and codomain $Y$ , where $Y$ is a metric space with distance $d$ . The sequence converges pointwise to $f:X\to Y$ if

\lim_{n\to\infty}f_n(x)=f(x) \qquad\text{for every }x\in X.

Thus, fixing $x$ produces an ordinary convergent sequence in $Y$ . This formulation applies, in particular, to functions taking values in the real numbers or complex numbers. (jirka.org)

Equivalently,

\forall x\in X\;\forall\varepsilon>0\; \exists N=N(x,\varepsilon)\; \forall n\ge N: \quad d(f_n(x),f(x))<\varepsilon.

The dependence of $N$ on $x$ is essential. Convergence may be fast at one point and arbitrarily slow near another. For real-valued functions, $d(f_n(x),f(x))$ becomes $|f_n(x)-f(x)|$ . (jirka.org)

The pointwise limit, if it exists, is unique in a metric space. Convergence on a subset $A\subseteq X$ means that the condition holds at every point of $A$ , without asserting anything about points outside $A$ . (jirka.org)

Examples and comparison with uniform convergence

Consider the polynomial functions

f_n(x)=x^{2n},\qquad x\in[-1,1].

For $|x|<1$ , the values tend to zero; at $x=-1$ and $x=1$ , they remain equal to one. Consequently,

f(x)= \begin{cases} 0,&|x|<1,\\ 1,&x=-1\text{ or }x=1. \end{cases}

Every $f_n$ is a continuous function, but the limiting function is discontinuous at the endpoints. Pointwise convergence therefore does not preserve continuity. (jirka.org)

Uniform convergence changes the order of the quantifiers:

\forall\varepsilon>0\;\exists N\; \forall x\in X\;\forall n\ge N: \quad d(f_n(x),f(x))<\varepsilon.

It implies pointwise convergence, but the converse fails. For the powers above, points arbitrarily close to either endpoint have values arbitrarily close to one, even though their limiting value is zero. Thus

\sup_{x\in[-1,1]}|f_n(x)-f(x)|=1

for every $n$ . On any smaller interval $[-a,a]$ , with $0<a<1$ , however, the error is at most $a^{2n}$ , which tends to zero uniformly. These estimates illustrate how the chosen domain affects convergence. (jirka.org)

Limits, integration, and differentiation

Pointwise convergence alone does not justify interchanging limiting operations. For continuous functions, uniform convergence supplies sufficient control to preserve continuity; pointwise convergence supplies no comparable general guarantee. Similar distinctions arise for integration and differentiation. (jirka.org)

For example, on $[0,1]$ , set

g_n(x)= \begin{cases} n,&0<x<1/n,\\ 0,&\text{otherwise}. \end{cases}

Direct calculation shows that $g_n(x)\to0$ at every fixed point: each positive $x$ eventually lies outside the shrinking interval, while $g_n(0)=0$ . Nevertheless,

\int_0^1g_n(x)\,dx=1, \qquad \int_0^1\lim_{n\to\infty}g_n(x)\,dx=0.

The increasing height compensates for the decreasing width, demonstrating the general failure of exchanging an integral and a pointwise limit. (jirka.org)

Even uniform convergence of differentiable functions need not permit exchanging differentiation and a limit. The functions

h_n(x)=\frac{\sin(nx)}{n}

converge uniformly to zero on $\mathbb R$ , since $|h_n(x)|\le1/n$ . Yet $h_n'(0)=1$ for every $n$ , whereas the derivative of the limiting zero function is zero. Appropriate differentiation theorems require additional hypotheses, commonly uniform convergence of the derivatives and convergence at one point. (ocw.mit.edu)

Measure-theoretic variants

In measure theory, convergence almost everywhere means pointwise convergence outside a null set. For real-valued measurable functions, an everywhere-existing pointwise limit is measurable. Unlike continuity, measurability is therefore preserved under pointwise sequential limits. (math.mit.edu)

The dominated convergence theorem provides conditions under which almost-everywhere convergence permits passage to the limit in a Lebesgue integral. If $f_n\to f$ almost everywhere and $|f_n|\le g$ almost everywhere for a fixed integrable function $g$ , then

\int |f_n-f|\,d\mu\longrightarrow0, \qquad \int f_n\,d\mu\longrightarrow\int f\,d\mu.

The dominating function supplies the control absent from pointwise convergence alone. (ocw.mit.edu)

Topological interpretation

If $Y$ is a topological space, pointwise convergence can be defined using convergence in $Y$ , without choosing a metric. Identify the set $Y^X$ of all functions $X\to Y$ with the Cartesian product $\prod_{x\in X}Y$ . Pointwise convergence is precisely convergence in the product topology. (math.mit.edu)

A basic neighborhood of $f$ restricts function values at only finitely many points:

\{g\in Y^X:g(x_i)\in U_i,\ i=1,\ldots,k\},

where each $U_i$ is an open neighborhood of $f(x_i)$ . This is also called the point-open topology. The equivalence extends from sequences to nets, allowing pointwise convergence to be treated within general topology. (math.mit.edu)