Deep Dive into Continuity

Continuity of Real Numbers

Completeness Property:

Every nonempty set of real numbers that is bounded above (or below) has a least upper bound (or greatest lower bound).

Monotone Convergence Theorem:

Every monotonic (non-decreasing or non-increasing) and bounded sequence of real numbers is convergent.

$$\begin{equation*} a_n \nearrow, \quad a_n \le M \implies \lim_{n \to +\infty} a_n = \sup_{n \ge 1} a_n \end{equation*}$$

Nested Interval Theorem:

Every sequence of nested closed and bounded intervals has a nonempty intersection.

$$\begin{equation*} [a_{n+1}, b_{n+1}] \subseteq [a_n, b_n] \implies \bigcap_{n \ge 1} [a_n, b_n] \ne \varnothing \end{equation*}$$

Closed Set

A subset $A \subseteq \mathbb{R}$ is called a closed set if it is closed under limits, that is:

$$\begin{equation*} \text{If } a_n \in A \text{ and } \lim_{n \to +\infty} a_n \text{ exists, then } \lim_{n \to +\infty} a_n \in A. \end{equation*}$$

Theorems

1. $K \subset \mathbb{R}$ is a bounded closed set if and only if every sequence in $K$ has a convergent subsequence whose limit also lies in $K$.

   $$\begin{equation*} K \text{ is bounded and closed } \iff \forall \{x_n\} \subset K, \exists \{x_{n_k}\} \text{ such that } x_{n_k} \to x^* \in K. \end{equation*}$$

2. Let $f$ be a continuous function on a nonempty bounded closed set $K \subset \mathbb{R}$. Then the image set $f(K)$ is also a nonempty bounded closed set. Therefore, $f$ attains both its maximum and minimum on $K$.

   $$\begin{equation*} \exists x_{\max}, x_{\min} \in K \quad \text{s.t.} \quad f(x_{\max}) = \max_{x \in K} f(x), \quad f(x_{\min}) = \min_{x \in K} f(x). \end{equation*}$$

Bolzano–Weierstrass

Every bounded sequence of real numbers has a convergent subsequence.

$$\begin{equation*} \text{If } (a_n) \text{ is bounded in } \mathbb{R}, \text{ then there exists a subsequence } (a_{n_k}) \text{ such that } \lim_{k \to +\infty} a_{n_k} \text{ exists.} \end{equation*}$$

Cauchy Convergence Criterion

A real sequence $\{a_n\}_{n \ge 1}$ converges if and only if it is a Cauchy sequence, that is:

$$\begin{equation*} \forall \varepsilon > 0, \, \exists N > 0 \text{ such that } \forall n \ge N, \, \forall p \ge 1, \, |a_{n+p} - a_n| < \varepsilon. \end{equation*}$$

Theorem (Banach Fixed Point Theorem / Contraction Mapping Theorem)

Let $A \subset \mathbb{R}$ be a non-empty closed set. Suppose $F: A \to \mathbb{R}$ satisfies:

1. $F(A) \subseteq A$;

2. (Contraction) There exists a constant $0 < \lambda < 1$ such that

   $$\begin{equation*} |F(x) - F(y)| \le \lambda |x - y|, \quad \forall x, y \in A. \end{equation*}$$

Then there exists a unique point $x^* \in A$ such that $F(x^*) = x^*$. Moreover, for any $x_0 \in A$ and any integer $n \ge 1$, we have

$$\begin{equation*} |F^n(x_0) - x^*| \le \frac{\lambda^n}{1 - \lambda} |F(x_0) - x_0|. \end{equation*}$$

Intermediate Value Property

If $f$ is continuous on the interval $[a,b]$ and $f(a) \ne f(b)$, then every real number between $f(a)$ and $f(b)$ is attained as a function value of $f$ on $[a,b]$.

Theorems

1. If a function $f$ is monotonic on an interval $I$, then $f$ is continuous if and only if $f(I)$ is an interval.

   $$\begin{equation*} f \text{ is continuous } \iff f(I) \text{ is an interval.} \end{equation*}$$

2. All elementary functions are continuous.

3. If $f$ is a continuous and injective (one-to-one) function on an interval $I$, then $f$ is strictly monotonic.

   $$\begin{equation*} \text{Continuous and injective } \Rightarrow \text{ Monotonic.} \end{equation*}$$

Uniform Continuity

A function $f$ is said to be uniformly continuous on a set $K$, if for every $\varepsilon > 0$, there exists a $\delta_\varepsilon > 0$ such that

$$\begin{equation*} \forall x, y \in K, \quad |x - y| < \delta_\varepsilon \implies |f(x) - f(y)| < \varepsilon. \end{equation*}$$

Theorems

1. If a function $f$ is continuous on a nonempty bounded closed set $K \subset \mathbb{R}$, then $f$ is uniformly continuous on $K$.