Linear Algebra
11.110 min read

Limits and Continuity in Multiple Variables

Before we can differentiate functions of multiple variables, we need to understand what it means for such a function to be continuous. The idea mirrors the single-variable case, but with a crucial twist: in Rn\mathbb{R}^n, there are infinitely many directions from which a point can be approached.

A function f:Rn→Rf: \mathbb{R}^n \to \mathbb{R} has limit LL at point a\mathbf{a} if f(x)f(\mathbf{x}) can be made arbitrarily close to LL by taking x\mathbf{x} sufficiently close to a\mathbf{a} — regardless of the direction of approach. Formally: for every ϵ>0\epsilon > 0, there exists δ>0\delta > 0 such that 0<∥x−a∥<δ0 < \|\mathbf{x} - \mathbf{a}\| < \delta implies ∣f(x)−L∣<ϵ|f(\mathbf{x}) - L| < \epsilon.

Continuity means the limit equals the function value: lim⁡x→af(x)=f(a)\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a}). All polynomials, rational functions (away from zeros of denominator), and compositions of continuous functions are continuous. The subtlety in multiple variables is that you cannot verify a limit by checking finitely many paths — the limit must hold along all paths simultaneously.

Formal View

Definition 11.1 — Limit in Multiple Variables
Let f:D⊆Rn→Rf: D \subseteq \mathbb{R}^n \to \mathbb{R} and let a\mathbf{a} be a limit point of DD. We say lim⁡x→af(x)=L\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L if for every ϵ>0\epsilon > 0 there exists δ>0\delta > 0 such that x∈D\mathbf{x} \in D and 0<∥x−a∥<δ0 < \|\mathbf{x} - \mathbf{a}\| < \delta imply ∣f(x)−L∣<ϵ|f(\mathbf{x}) - L| < \epsilon.
Definition 11.2 — Continuity
A function f:D⊆Rn→Rf: D \subseteq \mathbb{R}^n \to \mathbb{R} is continuous at a∈D\mathbf{a} \in D if lim⁡x→af(x)=f(a)\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a}). It is continuous on DD if it is continuous at every point of DD.
Remark 11.1
To show a limit does not exist in Rn\mathbb{R}^n, it suffices to find two paths to a\mathbf{a} along which ff has different limits. To show a limit does exist, path arguments are insufficient — one must use the ϵ\epsilon-δ\delta definition or squeeze theorem.

Why This Matters

Continuity is the baseline regularity assumption for virtually all theorems in multivariate calculus and optimization.

  • Ensuring optimization algorithms do not encounter discontinuous jumps in loss functions
  • Verifying that physical models (heat, fluid flow) are well-posed
  • Checking regularity conditions before applying the implicit function theorem

Quiz

Question 1

Which of the following is sufficient to prove that lim⁡x→0f(x)\lim_{\mathbf{x}\to\mathbf{0}} f(\mathbf{x}) does NOT exist?

Question 2

A function of two variables that is continuous along every straight line through a point is necessarily continuous at that point.

Common Mistakes

  • Checking only a few paths and concluding the limit exists — you need an ϵ\epsilon-δ\delta argument for existence.
  • Confusing "continuous at a point" with "limit exists at a point" — a function can have a limit at a point where it is not defined.
  • Forgetting that "distance" in Rn\mathbb{R}^n is the Euclidean norm, not a sum of absolute values (both work, but be consistent).