Limits and Continuity of Multivariate Functions
1.1 Examples of Multivariate Functions
An \(m\)-variable function is a mapping \(f : D \to \mathbb{R}\), where \(D\) is \(\mathbb{R}^m\) or a subset of it, and the function value \(y\) and independent variables \(x_1, x_2, \dots, x_m\) satisfy \(y = f(x_1, x_2, \dots, x_m)\).
The values of multiple variables \(y_1, \dots, y_n\) are determined by a set of independent variables \(x_1, \dots, x_m\):
\[ \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} f_1(x_1, \dots, x_m) \\ \vdots \\ f_n(x_1, \dots, x_m) \end{pmatrix}, \quad \text{i.e.,} \quad \begin{cases} y_1 = f_1(x_1, \dots, x_m), \\ \vdots \\ y_n = f_n(x_1, \dots, x_m). \end{cases} \]
1.2 Distance and Limits of Sequences in \(\mathbb{R}^m\)
A bivariate function \(d : \mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R}\) is called a distance on \(\mathbb{R}^m\) if it satisfies:
For any \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^m\), \(d(\mathbf{x}, \mathbf{y}) = d(\mathbf{y}, \mathbf{x}) \ge 0\);
For any \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^m\), \(d(\mathbf{x}, \mathbf{y}) = 0\) if and only if \(\mathbf{x} = \mathbf{y}\);
For any \(\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^m\), \(d(\mathbf{x}, \mathbf{z}) \le d(\mathbf{x}, \mathbf{y}) + d(\mathbf{y}, \mathbf{z})\).
For \(r > 0\), denote \[ B(\mathbf{x}, r) := \{\mathbf{y} \in \mathbb{R}^m \mid d(\mathbf{x}, \mathbf{y}) < r\}, \] which is called the open ball centered at \(\mathbf{x}\) with radius \(r\).
A distance \(d\) on \(\mathbb{R}^m\) is said to be translation-invariant if for any \(\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^m\),
\[ d(\mathbf{x} + \mathbf{z}, \mathbf{y} + \mathbf{z}) = d(\mathbf{x}, \mathbf{y}). \]
A function \(\|\cdot\| : \mathbb{R}^m \to \mathbb{R}\) is called a norm on \(\mathbb{R}^m\) if it satisfies: 1. For any \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^m\), \(\|\mathbf{x} + \mathbf{y}\| \le \|\mathbf{x}\| + \|\mathbf{y}\|\); 2. For any \(\mathbf{x} \in \mathbb{R}^m\), \(\|\mathbf{x}\| \ge 0\); \(\|\mathbf{x}\| = 0\) if and only if \(\mathbf{x} = \mathbf{0}\); 3. For any \(\mathbf{x} \in \mathbb{R}^m\) and any \(\lambda \in \mathbb{R}\), \(\|\lambda \mathbf{x}\| = |\lambda| \|\mathbf{x}\|\).
Given a norm \(\|\cdot\|\) in \(\mathbb{R}^n\) and a square matrix \(A\) of order \(n\), the matrix norm of \(A\) is defined as: \[ \begin{aligned} \|A\| = \max_{\|\mathbf{v}\|=1} \|A\mathbf{v}\|. \end{aligned} \]
Also \[ \begin{aligned} \|A\| = \max_{\mathbf{v}\neq 0} \frac{\|A\mathbf{v}\|}{\|\mathbf{v}\|}. \end{aligned} \]
Theorem 1.1
- For any matrix \(A\), the norm is always non-negative, and it is zero if and only if the matrix itself is the zero matrix.
\[ \begin{aligned} \|A\| \geq 0, \quad \text{and} \quad \|A\| = 0 \iff A = \mathbf{0} \end{aligned} \]
- Multiplying a matrix by a scalar \(\alpha\) scales the norm by the absolute value of that scalar.
\[ \begin{aligned} \|\alpha A\| = |\alpha| \cdot \|A\| \end{aligned} \]
- The norm of the sum of two matrices is less than or equal to the sum of their individual norms.
\[ \begin{aligned} \|A + B\| \leq \|A\| + \|B\| \end{aligned} \]
- For induced norms, the norm of a product of matrices is bounded by the product of their norms.
\[ \begin{aligned} \|AB\| \leq \|A\| \cdot \|B\| \end{aligned} \]
- By the very definition of the induced norm, the transformation of any vector \(\mathbf{v}\) by matrix \(A\) is bounded by the matrix norm:
\[ \begin{aligned} \|A\mathbf{v}\| \leq \|A\| \cdot \|\mathbf{v}\| \end{aligned} \]
A bivariate function \(\langle \cdot, \cdot \rangle : \mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R}\) is called an inner product on \(\mathbb{R}^m\) if it satisfies: 1. For any \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^m\), \(\langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle\); 2. For any \(\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^m\) and any \(\alpha, \beta \in \mathbb{R}\), \[ \langle \alpha \mathbf{x} + \beta \mathbf{y}, \mathbf{z} \rangle = \alpha \langle \mathbf{x}, \mathbf{z} \rangle + \beta \langle \mathbf{y}, \mathbf{z} \rangle; \] 3. For any \(\mathbf{x} \in \mathbb{R}^m\), \(\langle \mathbf{x}, \mathbf{x} \rangle \ge 0\); and \(\langle \mathbf{x}, \mathbf{x} \rangle = 0\) if and only if \(\mathbf{x} = \mathbf{0}\).
For any \(p \ge 1\), \[ \|\mathbf{x}\|_p := (|x^1|^p + \dots + |x^m|^p)^{\frac{1}{p}} \] defines a norm on \(\mathbb{R}^m\) ,which induces a distance on \(\mathbb{R}^m\): \[ d_p(\mathbf{x}, \mathbf{y}) := (|x^1 - y^1|^p + \dots + |x^m - y^m|^p)^{\frac{1}{p}}. \]
Theorem 1.2
\(\|\mathbf{x}\|_\infty := \max\{|x^1|, \dots, |x^m|\}\).
\(\|\mathbf{x}\|_\infty \le \|\mathbf{x}\|_p \le m^{\frac{1}{p}} \|\mathbf{x}\|_\infty\).
\(\frac{1}{m^{\frac{1}{q}}} \|\mathbf{x}\|_q \le \|\mathbf{x}\|_p \le m^{\frac{1}{p}} \|\mathbf{x}\|_q\).
Corollary 1.1 Any norm \(\|\cdot\|\) on \(\mathbb{R}^m\) is equivalent to the norm \(\|\cdot\|_\infty\),
Let \(\{\mathbf{x}_n\}_{n \ge 1}\) be a sequence in \(\mathbb{R}^m\). 1. The sequence \(\{\mathbf{x}_n\}_{n \ge 1}\) is said to be bounded if there exists \(M > 0\) such that for any positive integer \(n\), \(\|\mathbf{x}_n\| \le M\).
The sequence \(\{\mathbf{x}_n\}_{n \ge 1}\) is said to be convergent if there exists \(A \in \mathbb{R}^m\) such that for any \(\varepsilon > 0\), there exists a positive integer \(N(\varepsilon) > 0\) such that for any positive integer \(n\), whenever \(n \ge N(\varepsilon)\), we have \(\|\mathbf{x}_n - A\| < \varepsilon\). Such an \(A\) is called a limit of the sequence \(\{\mathbf{x}_n\}_{n \ge 1}\). We denote \(A = \lim\limits_{n \to +\infty} \mathbf{x}_n\).
The sequence \(\{\mathbf{x}_n\}_{n \ge 1}\) is said to be a Cauchy sequence if for any \(\varepsilon > 0\), there exists a positive integer \(N(\varepsilon) > 0\) such that for any positive integers \(p, q\), whenever \(p, q \ge N(\varepsilon)\), we have \(\|\mathbf{x}_p - \mathbf{x}_q\| < \varepsilon\).
Theorem 1.3 Let \(\{\mathbf{x}_n\}\) be a sequence in \(\mathbb{R}^m\), where \(\mathbf{x}_n = (x_n^1, \dots, x_n^m)^T\). Then:
- \(\{\mathbf{x}_n\}_{n \ge 1}\) is a bounded sequence if and only if for any \(1 \le k \le m\), the sequence \(\{x_n^k\}_{n \ge 1}\) is a bounded sequence.
- \(\{\mathbf{x}_n\}_{n \ge 1}\) is a Cauchy sequence if and only if for any \(1 \le k \le m\), the sequence \(\{x_n^k\}_{n \ge 1}\) is a Cauchy sequence.
- \(\lim\limits_{n \to +\infty} \mathbf{x}_n = A\) if and only if for any \(1 \le k \le m\), \(\lim\limits_{n \to +\infty} x_n^k = A_k\).
Theorem 1.4
- A sequence in \(\mathbb{R}^m\) is convergent if and only if it is a Cauchy sequence.
- Any bounded sequence in \(\mathbb{R}^m\) has a convergent subsequence.
A set \(E \subseteq \mathbb{R}^m\) is said to be a closed set if: for any sequence \(\{\mathbf{x}_n\}_{n \ge 1}\) in \(E\), if the sequence has a limit \(\lim\limits_{n \to +\infty} \mathbf{x}_n\), then this limit is also in \(E\).
1.3 Continuity of Multivariable Functions
Let \(E \subseteq \mathbb{R}^m\). A mapping \(f: E \to \mathbb{R}^p\) is said to be continuous at \(\mathbf{x}_0 \in E\), if for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any \(\mathbf{x} \in E\), as long as \[ \begin{aligned} \|\mathbf{x} - \mathbf{x}_0\| < \delta \end{aligned} \] the following inequality holds: \[ \begin{aligned} \|f(\mathbf{x}) - f(\mathbf{x}_0)\|_* < \epsilon \end{aligned} \] Here, \(\|\cdot\|\) and \(\|\cdot\|_*\) denote the norms in the domain space and the codomain space of \(f\), respectively.
- \(f\) is said to be continuous on \(E_0 \subseteq E\), if \(f\) is continuous at every point \(\mathbf{x}_0 \in E_0\).
- \(f\) is called a continuous mapping, if it is continuous on its entire domain \(E\).
Theorem 1.5 Any linear mapping on \(\mathbb{R}^m\) is continuous.
Theorem 1.6 Any bilinear mapping \(B: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^p\), i.e., \(\forall \alpha_1, \alpha_2, \beta_1, \beta_2 \in \mathbb{R}, \mathbf{x}_1, \mathbf{x}_2, \mathbf{x} \in \mathbb{R}^m, \mathbf{y}_1, \mathbf{y}_2, \mathbf{y} \in \mathbb{R}^n\) \[ \begin{aligned} B(\alpha_1 \mathbf{x}_1 + \alpha_2 \mathbf{x}_2, \mathbf{y}) &= \alpha_1 B(\mathbf{x}_1, \mathbf{y}) + \alpha_2 B(\mathbf{x}_2, \mathbf{y}), \\ B(\mathbf{x}, \beta_1 \mathbf{y}_1 + \beta_2 \mathbf{y}_2) &= \beta_1 B(\mathbf{x}, \mathbf{y}_1) + \beta_2 B(\mathbf{x}, \mathbf{y}_2), \end{aligned} \] is continuous.
Corollary 1.2 Any multilinear mapping is a continuous mapping.
Theorem 1.7 Vector addition, scalar multiplication, and the inner product in \(\mathbb{R}^m\), as well as real number multiplication, are all continuous mappings.
Theorem 1.8 Let \(E \subseteq \mathbb{R}^m\), \(F \subseteq \mathbb{R}^p\), and \(f: E \to \mathbb{R}^p\) be continuous at \(\mathbf{x}_0 \in E\). If \(g: F \to \mathbb{R}^q\) is continuous at \(\mathbf{y}_0 = f(\mathbf{x}_0) \in F\), then the composite mapping \[ \begin{aligned} g \circ f: E \cap f^{-1}(F) \to \mathbb{R}^q \end{aligned} \] is continuous at \(\mathbf{x}_0\).
Theorem 1.9 The determinant is a continuous function of the matrix; therefore, small perturbations of an invertible matrix remain invertible.
Theorem 1.10 Let \(E \subseteq \mathbb{R}^m\) and \(f: E \to \mathbb{R}^p\) be a continuous mapping. Then:
For any bounded closed subset \(K\) of \(E\), the image \(f(K)\) is also a bounded closed set.
If \(p = 1\), then for any non-empty bounded closed subset \(K\) of \(E\), \(f\) attains both a maximum value and a minimum value on \(K\).
Theorem 1.11 Let \(\langle \cdot, \cdot \rangle\) be an inner product on \(\mathbb{R}^n\), and let \(A\) be a symmetric matrix of order \(n\), i.e., \[ \begin{aligned} \langle A\mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{x}, A\mathbf{y} \rangle, \quad \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n. \end{aligned} \] There exist \(n\) orthonormal unit vectors \(\mathbf{v}_1, \dots, \mathbf{v}_n\) in \(\mathbb{R}^n\) and real numbers \(\lambda_1, \dots, \lambda_n\) such that \[ \begin{aligned} A\mathbf{v}_k = \lambda_k \mathbf{v}_k, \quad k = 1, \dots, n. \end{aligned} \]
Theorem 1.12 Let \(E \subseteq \mathbb{R}^m\) and \(f: E \to \mathbb{R}^p\) be a continuous mapping. For any bounded closed subset \(K\) of \(E\), \(f\) is uniformly continuous on \(K\).
This means that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any \(\mathbf{x} \in K\) and any \(\mathbf{y} \in E\), as long as \[ \begin{aligned} \|\mathbf{x} - \mathbf{y}\| < \delta \end{aligned} \] the following holds: \[ \begin{aligned} \|f(\mathbf{x}) - f(\mathbf{y})\|_* < \epsilon \end{aligned} \] Here, \(\|\cdot\|\) and \(\|\cdot\|_*\) denote the norms in the domain space and the codomain space of \(f\), respectively.
A set \(E \subseteq \mathbb{R}^m\) is called a path-connected set if, for any \(\mathbf{x}, \mathbf{y} \in E\), there exists a continuous mapping \(f : [0, 1] \to \mathbb{R}^m\) such that \[ \begin{aligned} f(0) = \mathbf{x}, \quad f(1) = \mathbf{y}, \quad \text{and } f(t) \in E \text{ for all } t \in [0, 1]. \end{aligned} \] Such a continuous mapping \(f\) is called a path from \(\mathbf{x}\) to \(\mathbf{y}\) in \(E\).
1.4 Limits of Multivariable Functions
Let \(E \subseteq \mathbb{R}^m\). A point \(\mathbf{x}_0 \in \mathbb{R}^m\) is called a cluster point (or accumulation point) of \(E\) if, for any \(\epsilon > 0\), there exists \(\mathbf{x} \in E\) such that \[ \begin{aligned} 0 < d(\mathbf{x}, \mathbf{x}_0) < \epsilon \end{aligned} \] If \(\mathbf{x}_0\) is both a cluster point of \(E \subseteq \mathbb{R}^m\) and a cluster point of its complement \(E^c = \mathbb{R}^m \setminus E\) (meaning every neighborhood of \(\mathbf{x}_0\) contains points from both \(E\) and \(\mathbb{R}^m \setminus E\)), then \(\mathbf{x}_0\) is called a boundary point of \(E\). The set of all boundary points of \(E\) is denoted as \(\partial E\) and is called the boundary set of \(E\).
Let \(E\) be a subset of the domain of mapping \(f\), and let \(\mathbf{x}_0 \in \mathbb{R}^m\) be a cluster point of \(E\). We say that \(\mathbf{A} \in \mathbb{R}^p\) is the limit of \(f(\mathbf{x})\) as \(\mathbf{x} \in E\) approaches \(\mathbf{x}_0\), denoted as \[ \begin{aligned} \lim_{E \ni \mathbf{x} \to \mathbf{x}_0} f(\mathbf{x}) = \mathbf{A} \end{aligned} \] if for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any \(\mathbf{x} \in E\), as long as \(0 < d(\mathbf{x}, \mathbf{x}_0) < \delta\), we have \[ \begin{aligned} d_2(f(\mathbf{x}), \mathbf{A}) < \epsilon \end{aligned} \] Here, \(d_2\) is the distance in the codomain space of \(f\).
- When \(E\) is the entire domain of mapping \(f\), \(\lim\limits_{E \ni \mathbf{x} \to \mathbf{x}_0} f(\mathbf{x})\) is simply written as \(\lim\limits_{\mathbf{x} \to \mathbf{x}_0} f(\mathbf{x})\).
Theorem 1.13 Let \(E\) be the domain of the composite function \(g \circ f\), with \(\lim\limits_{E \ni \mathbf{x} \to \mathbf{x}_0} f(\mathbf{x}) = \mathbf{y}_0\) and \(\lim\limits_{\mathbf{y} \to \mathbf{y}_0} g(\mathbf{y}) = A\). Define \[ \begin{aligned} \tilde{g}(\mathbf{y}) = \begin{cases} g(\mathbf{y}), & \mathbf{y} \neq \mathbf{y}_0; \\ A, & \mathbf{y} = \mathbf{y}_0. \end{cases} \end{aligned} \] Then \(\lim\limits_{\mathbf{x} \to \mathbf{x}_0} \tilde{g}(f(\mathbf{x})) = A\). In particular, if \(g\) is continuous at \(\mathbf{y}_0\), then \[ \begin{aligned} \lim_{\mathbf{x} \to \mathbf{x}_0} g(f(\mathbf{x})) = A. \end{aligned} \]