Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions

Let \(D\subseteq \mathbb R\), \(a\in D\) and let \(f,g:D\to\mathbb R\) be two real functions continuous at \(a\) and let \(\lambda\in\mathbb R\). Then the functions

\((1)\) \(f + g:D\to\mathbb R\)

\((2)\) \(\lambda f:D \to \mathbb R\)

\((3)\) \(f\cdot g:D \to \mathbb R\)

are continuous at \(a\).

Moreover, if \(g(a)\neq 0\), then the function

\((4)\) \(\frac fg:D' \to \mathbb R\) with the new domain \(D':=\{x\in D:g(x)\neq 0\}\)

is also continuous at \(a\).

Proofs: 1

Proofs: 1 2 3
Propositions: 4 5 6 7

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983