Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions

Throughout, $m \in \mathbb{N}$ with $m \ge 1$, and $\mathbb{R}^{m}$ carries the Euclidean norm $\lVert\cdot\rVert_2$ of The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ and The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$, whose induced metric is the published $d_2$ (Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page, $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it). A function into $\mathbb{R}^{m}$ is called vector-valued.

Continuity

Let $(X, d_X)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $A \subseteq X$ carry the restricted metric $d_A$ (Isometry, isometric embedding, and the subspace metric on a subset), let $f : A \to \mathbb{R}^{m}$ and let $a \in A$. Then $f$ is continuous at $a$ when

$(\forall \varepsilon > 0)\ (\exists \delta > 0)\ (\forall x \in A)\ \bigl[\ d_X(x,a) < \delta \ \Longrightarrow\ \lVert f(x) - f(a)\rVert_2 < \varepsilon\ \bigr],$

with $\varepsilon, \delta$ ranging over the positive reals, and continuous on $A$ when it is continuous at every point of $A$.

This is not a new notion, and that is the point of writing it down. Since $\lVert f(x)-f(a)\rVert_2 = d_2\bigl(f(x),f(a)\bigr)$ and $d_A$ is the restriction of $d_X$, the displayed condition is verbatim the condition of Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form for the map of metric spaces $f : (A,d_A) \to (\mathbb{R}^{m}, d_2)$. So every theorem about continuous maps of metric spaces applies to vector-valued functions with no translation, and this library has exactly one notion of continuity here. The same move was made once before, between the $\mathbb{R}$-native and the metric notions, in Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace; this item is that move one dimension up in the codomain.

The two cases used below are $X = \mathbb{R}$ with $d_{\mathbb{R}}(s,t) = |s-t|$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded) and $X = \mathbb{R}^{n}$ with $d_2$, for $n \ge 1$.

Limits, for a real domain

Let $A \subseteq \mathbb{R}$, let $f : A \to \mathbb{R}^{m}$, let $c$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$) and let $L \in \mathbb{R}^{m}$. We say $f(x)$ tends to $L$ as $x$ tends to $c$, and write $\lim_{x\to c} f(x) = L$, when

$(\forall \varepsilon > 0)\ (\exists \delta > 0)\ (\forall x \in A)\ \bigl[\ 0 < |x - c| < \delta \ \Longrightarrow\ \lVert f(x) - L\rVert_2 < \varepsilon\ \bigr].$

This is the condition of The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ with the absolute value in the codomain replaced by $\lVert\cdot\rVert_2$; as there, the puncture $0 < |x-c|$ is what makes $c$ a point the function need not be defined at, and the hypothesis that $c$ is a limit point of $A$ is what stops the condition from being satisfied vacuously.

The notation denotes: at most one $L$ satisfies the condition. Suppose $L$ and $L'$ both do and $L \ne L'$. Then $\varepsilon := \lVert L-L'\rVert_2 / 2 > 0$ by (N1) for $\lVert\cdot\rVert_2$ (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). Take $\delta$ and $\delta'$ for this $\varepsilon$ and put $\eta := \min\{\delta,\delta'\} > 0$. Since $c$ is a limit point of $A$ there is $x \in A$ with $0 < |x-c| < \eta$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), and then

$\lVert L - L'\rVert_2 \;\le\; \lVert L - f(x)\rVert_2 + \lVert f(x) - L'\rVert_2 \;<\; \varepsilon + \varepsilon \;=\; \lVert L-L'\rVert_2$

by (N3) and (N2), which trichotomy forbids. So $L = L'$.

Components

For $i < m$ define the $i$-th coordinate projection $\pi_i : \mathbb{R}^{m} \to \mathbb{R}$ by $\pi_i(y) := y_i = y(i)$, and for $f : A \to \mathbb{R}^{m}$ the $i$-th component $f_i := \pi_i \circ f$, a real-valued function on $A$.

Each $\pi_i$ is $1$-Lipschitz (Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction): for $y, z \in \mathbb{R}^{m}$,

$\bigl|\pi_i(y) - \pi_i(z)\bigr| \;=\; |y_i - z_i| \;\le\; \lVert y - z\rVert_2 \;=\; d_2(y,z),$

the middle inequality being $|w_i| \le \lVert w\rVert_2$ at $w := y-z$ (The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$ clause 3, or directly because $w_i^{2}$ is one term of the sum $\sum_{k<m}w_k^{2}$). Written in coordinates, $f(x)$ is the vector whose $i$-th coordinate is $f_i(x)$, and $f(x) = \sum_{i<m} f_i(x)\,e_i$ in the standard basis (The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$).

• The codomain $\mathbb{R}^{0}$ is excluded by the standing hypothesis $m \ge 1$, and nothing is lost. $\mathbb{R}^{0}$ has exactly one element, so every function into it is constant and every ball condition holds trivially; every such map is continuous and every limit is the unique point. That case is true, uninteresting, and not what this page is about. It is also outside the reach of $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, which defines the metrics only for $m \ge 1$.

• The domain may be any metric space, and this matters twice below. The derivative of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral needs a real domain, so it uses the limit clause; the companion page's function of two real variables needs the domain $\mathbb{R}^{2}$, so it uses the continuity clause. Both are instances of the same definition, and neither introduces a second notion.

• When $m = 1$ the codomain is $\mathbb{R}^{1}$, not $\mathbb{R}$. These are different sets, $\mathbb{R}^{1}$ being a set of functions $1 \to \mathbb{R}$. The map $\theta : \mathbb{R} \to \mathbb{R}^{1}$ sending $t$ to the function with value $t$ at $0$ is an isometric bijection for $d_2$ (Isometry, isometric embedding, and the subspace metric on a subset), and under it the notions above become those of Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point and The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$. Every comparison on this page between the vector-valued theory and the one-dimensional theory goes through that identification, stated explicitly each time.

• Convergence of sequences in $\mathbb{R}^{m}$ is not redefined here. It is Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$ for $(\mathbb{R}^{m},d_2)$, with balls as in Open ball, closed ball and sphere in a metric space, and its componentwise characterisation is For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm.