Limits

Let \(\require{physics} S\) be an open subset of \(\mathbb{R}^n\), and let \(\vb{f}\) be a function \[\begin{equation} \vb{f}: S \rightarrow \mathbb{R}^m: \vb{x} \mapsto \vb{f}(\vb{x}) \thinspace , \end{equation}\] in which we will designate vector-valued functions (i.e. \(m \leq 1\)) by a bold-face symbol. If we instead want to emphasize a scalar function (i.e. \(m=1\)), we will use an italic symbol.

The limit notation, in which \(\vb{x}\) approaches the interior point \(\vb{a}\) has two equivalent meanings: \[\begin{equation} \lim_{\vb{x} \to \vb{a}} \vb{f}(\vb{x}) = \vb{b} \iff \lim_{ ||\vb{x} - \vb{a}|| \to \vb{0}} ||\vb{f}(\vb{x}) - \vb{b}|| = 0 \thinspace . \end{equation}\] The function \(\vb{f}\) is called continuous at \(\vb{a}\) if \[\begin{equation} \lim_{\vb{x} \to \vb{a}} \vb{f}(\vb{x}) = \vb{f}(\vb{a}) \thinspace . \end{equation}\]