Continuity in R^n

Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}^m$, and let $a \in D$. The function $f$ is continuous at $a$ when, for each $\varepsilon > 0$, there is a $\delta > 0$ such that $\|f(x) - f(a)\| < \varepsilon$ for each $x \in D$ with $\|x - a\| < \delta$. Here $\|\cdot\|$ is the Euclidean norm of Euclidean space. The function $f$ is continuous on $D$ when it is continuous at each point of $D$.

Remarks

Let $a$ be a limit point of $D$. Then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$, with the limit of a function of several variables. A function into $\mathbb{R}^m$ is continuous if and only if each component function is continuous. Sums, products and compositions of continuous functions are continuous. So each polynomial in $x_1, \dots, x_n$ is continuous on $\mathbb{R}^n$. Continuity in each variable separately is weaker: see the separately continuous function that is not continuous.

Explanations

L1 human

A continuous function has no jumps: if you move the input a little, the output moves only a little. In the plane, "a little" means inside a small disc, so the condition must hold for moves in every direction.