[Understanding Analysis by Stephen Abbott] - [Chapter 4 Functional Limits and Continuity] - [4.3 Continuous Functions]

Definition 4.3.1 (Continuity) A function \(f : A \to \mathbb{R}\) is continuous at a point \(c \in A\) if, for all \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x - c| < \delta\) (and \(x \in A\)) it follows that \(|f(x) - f(c)| < \epsilon\)

If \(f\) is continuous at every point in the domain \(A\), then we say that \(f\) is continuous on \(A\)

Theorem 4.3.2 (Characterization of Continuity) Let \(f : A \to \mathbb{R}\), and let \(c \in A\). The function \(f\) is continuous at \(c\) if and only if any one of the following three conditions is met:

1. For all \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(|x - c| < \delta\) (and \(x \in A\)) implies \(|f(x) - f(c) < \epsilon|\)
2. For all \(V_\epsilon(f(c))\), there exists a \(V_\delta(c)\) with the property that \(x \in V_\delta(c)\) (and \(x \in A\)) implies \(f(x) \in V_\epsilon(f(c))\)
3. For all \((x_n) \to c\) (with \(x_n \in A\)), it follows that \(f(x_n) \to f(c)\)
if \(c\) is a limit point of \(A\), then the above conditions are equivalent to
4. \(\lim\limits_{x \to c}f(x) = f(c)\)

Corollary 4.3.3 (Criterion for Discontinuity). Let \(f : A \to \mathbb{R}\), and let \(c \in A\) be a limit point of \(A\). If there exists a sequence \((x_n) \subseteq A\) where \((x_n) \to c\) but such that \(f(x_n)\) does not converge to \(f(c)\), we may conclude that \(f\) is not continuous at \(c\).

Theorem 4.3.4 (Algebraic Continuity Theorem). Assume \(f : A \to \mathbb{R}\) and \(g : A \to \mathbb{R}\) are continuous at a point \(c \in A.\) Then,

1. \(kf(x)\) is continuous at \(c\) for all \(k \in \mathbb{R}\)
2. \(f(x) + g(x)\) is continuous at \(c\)
3. \(f(x)g(x)\) is continuous at \(c\)
4. \(f(x)/g(x)\) is continuous at \(c\), provided the quotient is defined.

Theorem 4.3.9 (Composition of Continuous Functions) Given \(f : A \to \mathbb{R}\) and \(g : B \to \mathbb{R}\), assume that the range \(f(A) = {f(x) : x \in A}\) is contained in the domain \(B\) so that the composition \(g \circ f(x) = g(f(x))\) is defined on A.

If \(f\) is continuous at \(c \in A\), and if \(g\) is continuous at \(f(c) \in B\), then \(g \circ f\) is continuous at \(c\)