The Timpson Lyceum · Real Analysis (2020)

Lecture 12 (Real Analysis 2020)

Overview

This lecture extends the idea of monotonicity from sequences to general functions and uses it to build a powerful continuity criterion: a monotone function whose image is an interval must be continuous. Combined with an earlier result, this gives a clean if-and-only-if characterization of continuity for monotone functions on intervals. We then use these tools to rigorously define $n$-th roots, rational powers, and prove they are continuous. Finally we introduce the limit point of a set and the limit of a function at a point, recovering the calculus notion of $\lim_{x \to x_0} f(x)$ in terms of sequences and recasting continuity in that language.

Key Concepts

Monotone functions. Let $f : D \to \mathbb{R}$.

• $f$ is monotonically increasing if for all $u, v \in D$, $u < v \implies f(u) \le f(v)$; strictly increasing if $u < v \implies f(u) < f(v)$.
• Decreasing versions flip the inequality. $f$ is monotone if it is increasing or decreasing. This generalizes monotone sequences, which are just functions on $\mathbb{N}$.

Image of a function. $f(D) = \{\, f(x) : x \in D \,\}$. It is an interval when it contains every point between any two of its points.

Monotone + interval image $\implies$ continuous. If $f : D \to \mathbb{R}$ is monotone and $f(D)$ is an interval, then $f$ is continuous. (The domain need not be an interval; only the image must be.) The proof exploits the interval property to find, for each $\varepsilon > 0$, a point of the image that monotonicity then pulls the sequence values past.

Characterization (corollary). If $f$ is monotone on an interval domain $I$, then $f \text{ is continuous} \iff f(I) \text{ is an interval}.$ The forward direction is the earlier theorem (a continuous function on an interval has interval image); the reverse is the result above.

Inverse functions. $f : E \to \mathbb{R}$ is one-to-one if $f(a) = f(b) \implies a = b$. A one-to-one $f$ has an inverse $f^{-1}$ sending each $y \in f(E)$ to the unique $x$ with $f(x) = y$, satisfying $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$.

Continuous-inverse theorem. If $f : I \to \mathbb{R}$ is strictly monotone on an interval $I$, then $f^{-1} : f(I) \to \mathbb{R}$ is continuous. (Reason: $f^{-1}$ is monotone with image $I$, an interval, so the criterion above applies.)

Roots and rational powers. Restricting $f(x) = x^n$ to $[0, \infty)$ makes it strictly increasing with interval image, so its inverse is defined and continuous; we set $x^{1/n} := f^{-1}(x)$. For $x > 0$ and a rational $r = m/n$ (with $n > 0$), define $x^r := (x^m)^{1/n}$, and $x^{-n} := 1/x^n$. These obey the usual laws ($x^a x^b = x^{a+b}$, equivalent fractions give equal powers).

Continuity of $x^r$. Since $x \mapsto x^m$ and $x \mapsto x^{1/n}$ are continuous and $x^r = (x^m)^{1/n}$ is their composition, $x^r$ is continuous on $(0, \infty)$ (compositions of continuous functions are continuous).

Limit point of a set. $x_0$ is a limit point of $D$ if there is a sequence in $D \setminus \{x_0\}$ converging to $x_0$. The limit point need not lie in $D$ (e.g. $0$ is a limit point of $(0,1)$). Every real number is a limit point of $\mathbb{Q}$ (because $\mathbb{Q}$ is dense in $\mathbb{R}$); likewise of the irrationals.

Limit of a function at a point. For $f : D \to \mathbb{R}$ and a limit point $x_0$ of $D$, $\lim_{x \to x_0} f(x) = L$ means: for every sequence $x_n \in D \setminus \{x_0\}$ with $x_n \to x_0$, the sequence $f(x_n) \to L$. This handles "holes" — $x_0$ need not be in $D$.

Continuity restated. For $x_0 \in D$, $f$ is continuous at $x_0$ iff $\lim_{x \to x_0} f(x) = f(x_0)$.

Limit laws (Theorem). If $\lim_{x \to x_0} f(x) = a$ and $\lim_{x \to x_0} g(x) = b$, then $\lim(f+g) = a+b$, $\lim(fg) = ab$, and $\lim(f/g) = a/b$ where $g \ne 0$. Composition: under matching domain/image conditions, $\lim_{x \to x_0} g(f(x)) = L$ when $f(x) \to y_0$ and $g(y) \to L$ as $y \to y_0$. All follow by reducing to the already-known sequence limit laws.

Worked Example

Prove $\displaystyle \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$ directly from the sequence definition. Take any sequence $x_n \to 1$ with $x_n \ne 1$, factor, cancel (valid since $x_n \ne 1$), and apply the sequence limit laws.

Since this holds for every such sequence, the function limit equals $2$, even though the function is undefined at $x = 1$.

Summary

• A monotone function whose image is an interval is automatically continuous; on an interval domain, continuity and "interval image" are equivalent for monotone functions.
• A strictly monotone function on an interval has a continuous inverse — the engine behind defining roots.
• Restricting $x^n$ to $[0,\infty)$ yields $x^{1/n}$; rational powers $x^{m/n} = (x^m)^{1/n}$ obey the usual exponent laws and are continuous (as compositions of continuous functions).
• $x_0$ is a limit point of $D$ if some sequence in $D \setminus \{x_0\}$ converges to it; it need not belong to $D$. Every real is a limit point of $\mathbb{Q}$.
• $\lim_{x \to x_0} f(x) = L$ means $f(x_n) \to L$ for every sequence $x_n \to x_0$ in $D \setminus \{x_0\}$; this lets us assign values at "holes."
• $f$ is continuous at $x_0$ exactly when $\lim_{x \to x_0} f(x) = f(x_0)$, and function limits inherit the sum, product, quotient, and composition laws from sequence limits.