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

Common mistakes and why they fail