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