Limits and continuity in several variables

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Same $\varepsilon$ – $\delta$ definition with $\norm{x - a}$ instead of $|x - a|$ . New subtlety: the point can be approached along infinitely many paths, and the limit must agree on all of them.

Formulas

$f(x,y) = \frac{xy}{x^2 + y^2}: \quad y = mx \implies f = \frac{m}{1 + m^2}$
no limit at the origin: it depends on the direction

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.