Continuity in Several Variables

The central idea

A function $f:D\subset\mathbb R^n\to\mathbb R^m$ is continuous at $\mathbf a\in D$ when $\lim_{\mathbf x\to\mathbf a}f(\mathbf x)=f(\mathbf a)$, with distance measured by a norm. For vector-valued output, continuity is equivalent to continuity of every component. Polynomials and valid compositions are continuous on their domains.

Definitions, hypotheses, and notation

All norms on finite-dimensional Euclidean space give the same notion of continuity, though the Euclidean norm is customary. Sequentially, continuity means every sequence of domain points approaching $\mathbf a$ has outputs approaching $f(\mathbf a)$. This form can disprove continuity by constructing two point sequences with incompatible limits.

Existence of partial derivatives does not imply continuity. Partial derivatives inspect coordinate directions, while continuity controls all directions jointly. Continuous first partial derivatives near a point do imply differentiability there and therefore continuity.

Conceptual meaning

Every path and every sufficiently close point must produce output close to the same value. Checking coordinate axes alone is insufficient because infinitely many curved approaches exist. Continuity couples the limit with the actual function value.

A dependable method and decision rule

1. Confirm the point lies in the domain.
2. Use continuity laws for standard formulas where denominators and radicals are valid.
3. At a piecewise join, estimate output error using $r=\|\mathbf x-\mathbf a\|$.
4. To disprove a limit, find two approaches with different results.
5. After finding a limit, compare it with the defined value.

Fully worked example

Graphical or geometric meaning

A small disk around the origin maps into a narrow vertical band around zero. The bound $|f|\le r$ controls all directions simultaneously, unlike a finite collection of path tests.

Common mistakes and why they fail

Verification and reasonableness checks

• Test notable paths as diagnostics, then use a uniform bound for proof.
• Verify denominators remain nonzero where a continuity law is invoked.
• For vector outputs, inspect each component.

One successful path proves nothing

Continuity of $f$ at $\mathbf a$ requires $f(\mathbf x)\to f(\mathbf a)$ as $\mathbf x$ approaches from every direction in the domain. Polynomials are continuous everywhere and rational functions where denominators are nonzero, so closure rules often settle routine cases. At a suspicious point, two paths with different limiting values prove discontinuity, but agreement along many paths does not prove continuity. A proof of existence needs a bound depending only on distance, such as $|f(\mathbf x)-L|\le C\|\mathbf x-\mathbf a\|^\alpha$. Polar coordinates can supply such a bound near the origin if angular factors remain bounded. At a domain boundary, approach only through points belonging to the domain and state that restriction explicitly.

Practice

1. Is $x^2+y^2$ continuous everywhere?
2. Where is $1/(x-y)$ continuous?
3. If a limit differs along two paths, is the function continuous?

Connections and next steps

• Limits of Multivariable Functions
• Functions of Several Variables
• Partial Derivatives