Continuity in R^n

Continuity at a Point

The previous tutorial defined limits of vector-valued functions by controlling the distance between an output and a proposed limiting vector. Continuity adds one requirement: at a point in the domain, the function’s value itself must be that limiting vector. This makes continuity a local condition on how nearby inputs affect the output, and it applies just as well when the domain is a subset of Euclidean space.

Let \(D\subseteq\mathbb{R}^m\), let \(f:D\to\mathbb{R}^n\), and fix \(a\in D\). Distances in the input space are measured in \(\mathbb{R}^m\), and distances in the output space are measured in \(\mathbb{R}^n\). Unless stated otherwise, we use the Euclidean norm in each space.

The implication includes \(x=a\), where the output distance is \(0\). Unlike the definition of a limit, continuity does not use a punctured neighborhood: it directly compares nearby outputs with \(f(a)\). The function \(f:D\to\mathbb{R}^n\) is continuous on \(D\) if it is continuous at every point of \(D\).

There is an important edge case. If \(a\) is isolated in \(D\), some open ball around \(a\) contains no other point of \(D\). Then every function on \(D\) is continuous at \(a\): for any \(\varepsilon>0\), choose a ball small enough that its only domain point is \(a\). The required output inequality is then just \(0<\varepsilon\).

Continuity and Limits

When \(a\) is an accumulation point of \(D\), continuity can be expressed using the limit from the previous tutorial. The point must be an accumulation point for that limit to be defined in the usual way.

Proof. Suppose first that \(f\) is continuous at \(a\). Given \(\varepsilon>0\), choose \(\delta>0\) from continuity. Whenever \(x\in D\) and \(0<\|x-a\|_2<\delta\), we also have \(\|x-a\|_2<\delta\), so \[ \|f(x)-f(a)\|_2<\varepsilon. \] This is the definition of the limit \(f(a)\).

Conversely, suppose \(\lim_{x\to a,\ x\in D}f(x)=f(a)\). Given \(\varepsilon>0\), choose \(\delta>0\) so that \[ x\in D,\quad 0<\|x-a\|_2<\delta \quad\Longrightarrow\quad \|f(x)-f(a)\|_2<\varepsilon. \] If \(x\in D\) and \(\|x-a\|_2<\delta\), then either \(x=a\), in which case \(\|f(x)-f(a)\|_2=0<\varepsilon\), or \(x\neq a\), in which case the displayed implication applies. Thus \(f\) is continuous at \(a\). \(\square\)

A Sequential Criterion for Continuity

Continuity can also be tested by looking at sequences in the domain. The criterion differs slightly from the sequential criterion for limits: sequences are allowed to equal \(a\), including infinitely often. This is useful because continuity concerns all domain points near \(a\), not only points distinct from it.

Proof. Suppose \(f\) is continuous at \(a\), and let \((x_k)\) be a sequence in \(D\) converging to \(a\). Given \(\varepsilon>0\), choose \(\delta>0\) from the definition of continuity. Since \(x_k\to a\), there is an integer \(K\) such that \(k\geq K\) implies \(\|x_k-a\|_2<\delta\). Hence, for every \(k\geq K\), \[ \|f(x_k)-f(a)\|_2<\varepsilon. \] Thus \(f(x_k)\to f(a)\).

Conversely, suppose the sequential condition holds, but \(f\) is not continuous at \(a\). The failure of continuity means that there is an \(\varepsilon_0>0\) such that for every \(\delta>0\), there is an \(x\in D\) with \[ \|x-a\|_2<\delta \quad\text{and}\quad \|f(x)-f(a)\|_2\geq\varepsilon_0. \] For each positive integer \(k\), apply this statement with \(\delta=1/k\) to choose \(x_k\in D\) such that \[ \|x_k-a\|_2<\frac{1}{k} \quad\text{and}\quad \|f(x_k)-f(a)\|_2\geq\varepsilon_0. \] The first inequality shows \(x_k\to a\). By the assumed sequential condition, \(f(x_k)\to f(a)\). But the second inequality holds for every \(k\), so the output sequence cannot converge to \(f(a)\). This contradiction proves continuity at \(a\). \(\square\)

A single sequence with outputs that fail to converge to \(f(a)\) is enough to disprove continuity. To prove continuity using sequences, however, one must establish the required output convergence for every sequence in the domain that approaches \(a\).

Continuity of Compositions

Continuity is stable under composition: if one function sends nearby inputs to nearby intermediate values, and a second function sends those intermediate values to nearby outputs, then the combined function is continuous. The neighborhoods used for the two functions need not have the same radius, so the proof chooses them in sequence.

Proof. Let \(\varepsilon>0\). By continuity of \(g\) at \(f(a)\), there is an \(\eta>0\) such that for every \(y\in E\), \[ \|y-f(a)\|_2<\eta \quad\Longrightarrow\quad \|g(y)-g(f(a))\|_2<\varepsilon. \] By continuity of \(f\) at \(a\), there is a \(\delta>0\) such that for every \(x\in D\), \[ \|x-a\|_2<\delta \quad\Longrightarrow\quad \|f(x)-f(a)\|_2<\eta. \] For such an \(x\), the value \(f(x)\) belongs to \(E\), so the implication for \(g\) applies with \(y=f(x)\). Therefore, \[ \|g(f(x))-g(f(a))\|_2<\varepsilon. \] Since \((g\circ f)(a)=g(f(a))\), this is exactly continuity of \(g\circ f\) at \(a\). \(\square\)

What Continuity Does and Does Not Say

Continuity is local: continuity at \(a\) controls outputs for inputs sufficiently close to that particular point. The suitable \(\delta\) may depend on both \(a\) and \(\varepsilon\). Continuity at every point of a domain therefore does not, by itself, provide one common \(\delta\) that works throughout the domain.

It is also important to distinguish continuity from the existence of a limit. At an accumulation point, continuity means that the limit exists and equals the assigned value \(f(a)\). If a limit exists but \(f(a)\) is assigned a different value, the function is not continuous at \(a\). At an isolated domain point, by contrast, continuity holds automatically, even though the limit definition from the previous tutorial does not apply there.

The Sequential Criterion is particularly effective for detecting failure: one sequence approaching \(a\) whose outputs do not approach \(f(a)\) settles the matter. For proving continuity, direct estimates can be shorter, while the composition theorem lets continuity be assembled from simpler functions. The next topic develops a further practical method for functions whose outputs have several coordinates.