Limit

Open ball

The set of points \(B(\mathbf{a},r)\) such that their distance to the given point \(\mathbf{a}\) is strictly less than the given distance \(r\), i.e.,

\[B(\mathbf{a}, r)=\{ \mathbf{x}\in\mathbb{R}^n: \| \mathbf{x}-\mathbf{a} \| \lt r \}\]

Closed Ball

The set of points \(\bar{B}(\mathbf{a}, r)\) such that their distance to the given point \(\mathbf{a}\) is equal to or less than the given distance \(r\), i.e.,

\[\bar{B}(\mathbf{a}, r)=\{ \mathbf{x}\in\mathbb{R}^n: \| \mathbf{x}-\mathbf{a} \| \le r \}\]

Boundary Points

Suppose set \(X\) is a subset of \(\mathbb{R}^n\), and some point \(\mathbf{a}\in X\). If any open ball centered at \(\mathbf{a}\) simultaneously contains points that belongs to \(X\) and points that do not belong to \(X\), then \(\mathbf{a}\) is called a boundary point of \(X\).

Interior Points

Suppose \(X\) is a subset of \(\mathbb{R}^n\) and some point \(\mathbf{a}\in X\). If there exists some \(r\gt 0\) such that \(B(\mathbf{a}, r)\subset X\), then \(\mathbf{a}\) is called an interior point of \(X\).

Open Set

Suppose \(X\subset\mathbb{R}^n\), if any \(x\in X\) is an interior point of \(X\), then \(X\) is an open set.

Closed Set

Suppose \(X\subset\mathbb{R}^n\), if its complement \(\complement_U X\) is an open set, then \(X\) is a closed set.

Limit Points

Suppose \(X\subset\mathbb{R}^n\), if some point \(\mathbf{a}\in\mathbb{R}^n\) satisfies that for all \(\delta\gt 0\), there always exists some point \(\mathbf{x}\in X\cap B(\mathbf{a}, \delta)\) and \(\mathbf{x}\ne\mathbf{a}\), then \(\mathbf{a}\) is a limit point of \(X\), or an accumulation of \(X\).

Isolated Points

If \(\mathbf{x}\in X\) is not a limit point, then it’s an isolated point.

For vector-valued functions, suppose function \(\mathbf{f}:X\mapsto\mathbb{R}^m\) where \(X\subset\mathbb{R}^n\) and \(\mathbf{a}\) is a limit point of \(X\). We say \(\mathbf{L}\) to be the limit of \(\mathbf{f}\) at \(\mathbf{a}\), if and only if for any \(\epsilon \gt 0\), there exists some \(\delta \gt 0\) such that when \(\mathbf{x}\in X\) and \(0\lt \| \mathbf{x}-\mathbf{a} \| \lt \delta\), we have \(\| \mathbf{f}(\mathbf{a}) - \mathbf{L} \| \lt \epsilon\). We denote

\[\lim_{\mathbf{x}\to \mathbf{a}} \mathbf{f}(\mathbf{x})=\mathbf{L}\]

Proposition. Let \( \mathbf{f}:X\mapsto\mathbb{R}^{m} \) be a function where \( X\subset\mathbb{R}^{n} \). Let \( \mathbf{a} \) be a limit point of \( X \). If the limit of \( \mathbf{f} \) at \( \mathbf{a} \) exists, then this limit is uniquely determined.