Sequences of Functions

16 Sequences of Functions

We have looked at sequences and series of real numbers. Our next topic, from Section 4.5 in the textbook, is sequences of functions.

Let $I$ be some subset of $R$ . Suppose that for each $n \in N$ , $f_{n} (x)$ is a function from $I$ to $R$ . Then ${f_{n}}_{n = 1}^{\infty}$ is a sequence of functions on $I$ . Given such a sequence of functions, for each $x \in I$ , we get a sequence of real numbers ${f_{n} (x)}_{n = 1}^{\infty}$ , and we can ask about the convergence of that sequence. If $lim_{n \to \infty} f_{n} (x)$ exists for each $x \in I$ , then we can define a function $f$ on $I$ by letting $f (x) = lim_{n \to \infty} f_{n} (x)$ . We then say that the sequence of functions, ${f_{n}}_{n = 1}^{\infty}$ , is is pointwise convergent on $I$ and that it converges pointwise to $f$ .

For example, let $I = [0, \infty)$ and let $f_{n} (x) = \frac{1}{1 + n x}$ . Looking at $x = 0$ , $f_{n} (0) = 1$ for all $n$ , so $lim_{n \to \infty} f_{n} (0) = 0$ . For any $x > 0$ , $lim_{n \to \infty} (1 + n x) = + \infty$ and $lim_{n \to \infty} f_{n} (x) = 0$ . We see that the sequence of functions ${f_{n}}_{n = 1}^{\infty}$ converges pointwise on $[0, \infty)$ to the function $f (x) = {\begin{cases} 0 & if x = 0 \\ 1 & if x > 0 \end{cases}$

Note that we have a sequence of continuous functions that converges to a discontinuous function. In fact, for many purposes, pointwise convergence is not a "strong" enough form of convergence. A stronger form is given by "uniform convergence." Uniform convergence relates to pointwise convergence similarly to the way uniform continuity relates to continuity. That is, the difference between pointwise convergence and uniform convergence is the order in which quantifiers are applied. Note that ${f_{n}}_{n = 1}^{\infty}$ converges pointwise on $I$ to $f$ if for every $ε > 0$ and every $x \in I$ , there is a $N \in N$ (depending on both $ε$ and $x$ ) such that for all $n \geq N$ , $| f_{n} (x) - f (x) | < ε$ . For uniform convergence, given an $ε > 0$ there must exist an $N$ , depending on $ε$ , only that works for all $x \in I$ .

Definition Let ${f_{n}}_{n = 1}^{\infty}$ be a sequence of functions defined on the subset $I$ of $R$ , and let $f$ be a function defined on $I$ . We say that ${f_{n}}_{n = 1}^{\infty}$ converges uniformly to $f$ on $I$ if for all $ε > 0$ , there is an $N \in N$ such that for all $x \in I$ and all $n \geq N$ , $| f_{n} (x) - f (x) | < ε$ . We say that ${f_{n}}_{n = 1}^{\infty}$ is uniformly convergent.

Uniform convergence has many nice properties that pointwise convergence lacks. For example, the uniform limit of continuous functions is continuous.

Theorem: Suppose that ${f_{n}}_{n = 1}^{\infty}$ is a sequence of continuous functions on an interval $I$ , and that ${f_{n}}_{n = 1}^{\infty}$ converges uniformly to $f$ on $I$ . Then $f$ is continuous on $I$ .

Similarly, the integral of a pointwise limit of functions $f_{n}$ on $[a, b]$ is not necessarily the limit of the integrals $\int_{a}^{b} f_{n}$ , even if that limit exists. But

Theorem: Suppose that ${f_{n}}_{n = 1}^{\infty}$ is a sequence of differentiable functions on the interval $[a, b]$ and that ${f_{n}}_{n = 1}^{\infty}$ converges uniformly to $f$ on $[a, b]$ . Then $f$ is integrable on $[a, b]$ and $\int_{a}^{b} f = lim_{n \to \infty} \int_{a}^{b} f_{n}$

And for derivatives, it is not necessarily true that the derivative of a pointwise limit is the limit of the derivatives. However, that will be the case if the derivative functions converge uniformly.

Theorem: Suppose that ${f_{n}}_{n = 1}^{\infty}$ is a sequence of continuous functions on the interval $[a, b]$ and that ${f_{n}}_{n = 1}^{\infty}$ converges pointwise to $f$ on $[a, b]$ . Suppose also that the sequence of derivative functions, ${f_{n}^{'}}_{n = 1}^{\infty}$ , converges uniformly on $[a, b]$ . Then $f$ is differentiable and ${f_{n}^{'}}_{n = 1}^{\infty}$ converges to $f^{'}$ on $[a, b]$ . That is, for all $x \in [a, b]$ , $f^{'} (x) = lim_{n \to \infty} f_{n}^{'} (x)$

Kevin Mitchell, one of the authors of our textbook, has a web page with animations of some of the sequences of functions from the examples and exercises in Section 4.5. The link is here:

Foundations of Analysis: Sequences of Functions

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