Limits and Continuity

The third page of the analysis spine. We define limits and continuity of real functions in the $\epsilon$–$\delta$ language, prove the two theorems that make continuity powerful on closed bounded intervals — the intermediate value and extreme value theorems — and reconcile the analytic definition with the topological one used in topology-manifolds.md.

References: Rudin, Principles of Mathematical Analysis, ch. 4; Tao, Analysis I.

1. Limits of Functions

Let $f:E\to \mathbb{R}$ with $E\subseteq \mathbb{R}$, and let $c$ be a limit point of $E$ (every neighbourhood of $c$ meets $E\setminus \{c\}$). We say

$$\underset{x\to c}{\lim }f(x)=L\text{iff}\forall \epsilon >0\exists \delta >0\forall x\in E:0<|x-c|<\delta ⟹|f(x)-L|<\epsilon .$$

The value $f(c)$ (if defined) is irrelevant to the limit.

Sequential characterization. ${\lim }_{x\to c}f(x)=L$ iff $f(x_n)\to L$ for every sequence $x_n\in E\setminus \{c\}$ with $x_n\to c$. This bridges to sequences-series.md and is the easiest way to disprove a limit (exhibit two sequences with different limits).

The algebra of limits carries over from sequences: limits respect sums, products, and quotients (denominator limit $\ne 0$). One-sided limits ${\lim }_{x\to c^{\pm }}$ are defined by restricting to $x>c$ or $x<c$.

2. Continuity

$f:E\to \mathbb{R}$ is continuous at $c\in E$ if for every $\epsilon >0$ there is $\delta >0$ such that $x\in E$ and $|x-c|<\delta$ imply $|f(x)-f(c)|<\epsilon$. Equivalently (when $c$ is a limit point) ${\lim }_{x\to c}f(x)=f(c)$. $f$ is continuous on $E$ if continuous at every point of $E$.

Sums, products, quotients (nonzero denominator), and compositions of continuous functions are continuous. Polynomials, $\exp$, $\sin$, $\cos$, and $|\cdot |$ are continuous everywhere; $\sqrt{\cdot }$ on $[0,\infty )$.

2.1 The topological definition

Theorem. $f:\mathbb{R}\to \mathbb{R}$ is continuous iff the preimage $f^{-1}(U)$ of every open set $U$ is open.

This is exactly the definition of continuity used for general topological spaces in topology-manifolds.md §1.2. The $\epsilon$–$\delta$ formulation is the metric specialization: open balls $(f(c)-\epsilon ,f(c)+\epsilon )$ pull back to open neighbourhoods of $c$. Keeping both views in mind is essential once we pass to ${\mathbb{R}}^n$ and manifolds — the topological definition is what survives the generalization.

3. The Intermediate Value Theorem

Intermediate Value Theorem (IVT). If $f:[a,b]\to \mathbb{R}$ is continuous and $y$ lies between $f(a)$ and $f(b)$, then $f(c)=y$ for some $c\in [a,b]$.

Proof. Assume $f(a)<y<f(b)$ and let $S=\{x\in [a,b]:f(x)<y\}$. It is nonempty ($a\in S$) and bounded, so $c=\sup S$ exists by completeness (real-numbers.md). Continuity rules out both $f(c)<y$ (we could move right and stay in $S$) and $f(c)>y$ (points just left of $c$ would have $f\ge y$, contradicting $c=\sup S$). Hence $f(c)=y$. $\square$

IVT is the analytic content behind "a connected set has connected image" and is used for root-finding (bisection) and existence proofs throughout calculus.

4. Compactness and the Extreme Value Theorem

A set $K\subseteq \mathbb{R}$ is compact iff it is closed and bounded (Heine–Borel; the general statement is in metric-spaces.md). The two facts we need:

• A continuous image of a compact set is compact.
• A nonempty compact subset of $\mathbb{R}$ contains its sup and inf.

Extreme Value Theorem (EVT). A continuous function on a closed bounded interval $[a,b]$ attains a maximum and a minimum.

Proof. $f([a,b])$ is compact, hence bounded and contains $\sup f([a,b])$ and $\inf f([a,b])$. The points where these are attained are the extremizers. (Concretely: take $x_n$ with $f(x_n)\to \sup f$; by Bolzano–Weierstrass a subsequence $x_{n_k}\to x^{\ast }\in [a,b]$, and continuity gives $f(x^{\ast })=\sup f$.) $\square$

EVT is the existence guarantee behind optimization and is invoked in the proof of the Mean Value Theorem (differentiation.md).

5. Uniform Continuity

$f:E\to \mathbb{R}$ is uniformly continuous if

$$\forall \epsilon >0\exists \delta >0\forall x,y\in E:|x-y|<\delta ⟹|f(x)-f(y)|<\epsilon ,$$

with a single $\delta$ that works everywhere (contrast ordinary continuity, where $\delta$ may depend on the point). E.g. $x\mapsto x^2$ is continuous but not uniformly continuous on $\mathbb{R}$; $x\mapsto 1/x$ fails uniform continuity on $(0,1)$.

Heine–Cantor Theorem. A continuous function on a compact set is uniformly continuous.

Proof sketch. If not, there are $\epsilon >0$ and points $x_n,y_n$ with $|x_n-y_n|\to 0$ but $|f(x_n)-f(y_n)|\ge \epsilon$. A convergent subsequence $x_{n_k}\to x^{\ast }$ (Bolzano–Weierstrass) forces $y_{n_k}\to x^{\ast }$ too, and continuity at $x^{\ast }$ contradicts the gap. $\square$

Uniform continuity is precisely the hypothesis that makes a continuous function Riemann integrable on $[a,b]$ (riemann-integral.md §3), so this theorem is a key structural input downstream.

6. Discontinuities and Monotone Functions

A function can fail continuity at $c$ via a jump (both one-sided limits exist but differ or miss $f(c)$) or an essential discontinuity (a one-sided limit fails to exist, e.g. $\sin (1/x)$ at $0$). A monotone function on an interval has at most countably many discontinuities, all jumps — a fact that reappears when classifying Riemann-integrable functions.

7. Where this page is used

• EVT $\to$ Rolle's theorem and the Mean Value Theorem in differentiation.md.
• Uniform continuity on compacts $\to$ integrability of continuous functions in riemann-integral.md.
• Topological continuity is the version that generalizes to ${\mathbb{R}}^n$ and manifolds; the multivariable theory begins in metric-spaces.md and multivariable.md.

Next: Differentiation.