# Introduction

Analysis, at its core, is the study of limiting processes. Every major construction in the subject — derivatives, integrals, [power series](/page/Power%20Series), Fourier expansions — is defined as a limit of some kind. A derivative is a limit of difference quotients; a [Riemann integral](/page/Riemann%20Integral) is a limit of sums; a power series is a limit of partial sums of [functions](/page/Function). The central difficulty of analysis is that these limiting processes do not always behave as one might hope: the limit of a sequence of continuous functions need not be continuous, the integral of a limit need not equal the limit of the integrals, and the derivative of a limit need not equal the limit of the derivatives.

This tension — between the desire to pass [limits](/page/Limit) through operations and the fact that doing so is not always valid — drives the entire theory developed in these notes. The resolution will come in stages: first, by identifying *modes of convergence* strong enough to guarantee that limits commute with the operations we care about; later, by developing abstract frameworks ([metric spaces](/page/Metric%20Space), [topological](/page/Topology) spaces) in which the relevant notions of convergence, [continuity](/page/Continuity), and compactness can be formulated cleanly and applied to function spaces themselves.

## The Fundamental Problem

To see why care is needed, consider the most natural notion of convergence for a sequence of functions.

[definition: Pointwise Convergence] Let $E$ be a set and let $f_n: E \to \mathbb{R}$ be a sequence of functions. We say $(f_n)$ **converges pointwise** to $f: E \to \mathbb{R}$ if for every $x \in E$, \begin{align*} \lim_{n \to \infty} f_n(x) = f(x). \end{align*} That is, for each fixed $x \in E$ and each $\varepsilon > 0$, there exists $N = N(x, \varepsilon) \in \mathbb{N}$ such that $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$. [/definition]

The notation $N = N(x, \varepsilon)$ is deliberate — it signals that the rate of convergence may vary wildly from point to point. This dependence on $x$ is the source of every pathology that follows.

[example: Continuity Destroyed by Pointwise Limits] Define $f_n: [0, 1] \to \mathbb{R}$ by $f_n(x) = x^n$. Each $f_n$ is continuous (indeed, a polynomial), and the sequence converges pointwise to \begin{align*} f(x) = \begin{cases} 0 &amp; \text{if } 0 \le x &lt; 1, \\ 1 &amp; \text{if } x = 1. \end{cases} \end{align*} The limit $f$ is discontinuous at $x = 1$. The mechanism is clear: at any fixed $x &lt; 1$, the convergence $x^n \to 0$ is exponentially fast — at $x = 1/2$, for instance, $f_n(1/2) = 2^{-n} &lt; \varepsilon$ once $n > \log_2(1/\varepsilon)$. But as $x$ approaches $1$, the convergence slows without bound. At $x = 1 - 1/n$, we have $f_n(x) = (1 - 1/n)^n \to 1/e \approx 0.368$, so $f_n(x)$ is still far from the limit when $n$ is of order $1/(1-x)$. No single $N$ controls the error uniformly across the interval — the convergence is fast on $[0, 1/2]$, slow near $1$, and fails entirely at $1$. [/example]

This example is not an isolated curiosity. The failure of pointwise convergence to preserve continuity is the rule rather than the exception. Far worse pathologies exist: pointwise limits of continuous functions can be discontinuous at every point of a dense set (though not at *every* point — the set of continuity points is always a dense $G_\delta$ set, by the [Baire Category Theorem](/theorems/630), a result we will encounter when we study completeness in metric spaces).

The situation for integrals is, if anything, more dramatic.

[example: Integration Destroyed by Pointwise Limits] Define $f_n: [0, 1] \to \mathbb{R}$ by \begin{align*} f_n(x) = \begin{cases} n^2 x &amp; \text{if } 0 \le x \le 1/n, \\ n^2(2/n - x) &amp; \text{if } 1/n &lt; x \le 2/n, \\ 0 &amp; \text{if } x > 2/n. \end{cases} \end{align*} Each $f_n$ is a triangular "spike" of height $n$ and base width $2/n$. The area under each spike is $\int_0^1 f_n = \frac{1}{2} \cdot \frac{2}{n} \cdot n = 1$ for every $n$. But $f_n(x) \to 0$ pointwise for every $x > 0$ (since $f_n(x) = 0$ once $n > 2/x$), so the pointwise limit is $f \equiv 0$. The limit of the integrals is $1$, but the integral of the limit is $0$: \begin{align*} \lim_{n \to \infty} \int_0^1 f_n \, dx = 1 \neq 0 = \int_0^1 \lim_{n \to \infty} f_n(x) \, dx. \end{align*} The mass concentrates at the origin and escapes to infinity in height, invisible to the pointwise limit. This is not a deficiency of the Riemann integral — the same failure occurs for Lebesgue integrals. The problem lies entirely with pointwise convergence. [/example]

## The Resolution: Uniform Convergence

Both examples above fail for the same structural reason: the convergence $f_n(x) \to f(x)$ is fast at some points and arbitrarily slow at others. The natural fix is to demand that the convergence happen at the same rate everywhere — that is, to require the existence of a single $N$ that works simultaneously for all $x$. This is the idea of **uniform convergence**, which we develop in full in the next chapter. The key results are:

- The **Uniform Limit Theorem**: if $f_n \to f$ uniformly and each $f_n$ is continuous, then so is $f$. - **Interchange of limits and integrals**: if $f_n \to f$ uniformly on $[a, b]$ and each $f_n$ is Riemann integrable, then $\int f_n \to \int f$. - **Interchange of limits and derivatives**: this requires not uniform convergence of $f_n$ but uniform convergence of $f_n'$ — a subtle but important distinction, since [differentiation](/page/Derivative) amplifies oscillations.

Uniform convergence is the right tool for many problems, particularly the convergence of power series within their radius of convergence. However, it is far from the end of the story.

## Beyond Uniform Convergence

Uniform convergence is a strong condition, and in many important situations it is too strong. The spike example above cannot be "fixed" by any mode of convergence on $\mathbb{R}$-valued functions alone — the issue is that $L^p$ norms are not controlled by pointwise limits. This leads, in more advanced courses, to the Lebesgue theory of integration and the notion of convergence in $L^p$ spaces, where the [Dominated Convergence Theorem](/theorems/4) and [Monotone Convergence Theorem](/theorems/509) provide the correct interchange results.

Even within the scope of these notes, we will need to go beyond concrete function spaces on $\mathbb{R}$. Many of the results about uniform convergence are most naturally stated in terms of the **uniform norm** $\|f\|_\infty = \sup_x |f(x)|$, which makes the space of bounded functions into a metric space. The Cauchy criterion for uniform convergence is precisely the assertion that this metric space is *complete*. To make such statements precise, and to handle convergence in more general settings (function spaces, spaces of [sequences](/page/Sequence), spaces of measures), we need the abstract theory of **metric spaces** and **topological spaces**.

## Overview of These Notes

The notes are organised around two interlocking themes: the theory of convergence for sequences and [series](/page/Series) of functions, and the abstract framework of metric and topological spaces that underpins it.