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For a multivariable function, lim_(x,y)→(a,b) f(x,y) = L if for every ε > 0 there exists δ > 0 such that |f(x,y) − L| < ε whenever 0 < √[(x−a)² + (y−b)²] < δ. Continuity requires the limit to exist and equal f(a, b). Multiple paths to a point complicate convergence analysis.
From single-variable limits, you know that lim_{x→a} f(x) = L means f(x) can be made arbitrarily close to L by taking x close enough to a — from either side. In one dimension, "from either side" covers all possible directions of approach. In two dimensions, the situation is fundamentally harder: there are infinitely many paths through (a, b) — lines at every angle, parabolas, spirals, spirals that spiral inward — and the limit lim_{(x,y)→(a,b)} f(x,y) = L must hold along every single one of them simultaneously.
This is the key new difficulty. If even one path to (a, b) gives a different limiting value, or no limit at all, then the two-variable limit does not exist. The path test exploits this: substitute y = mx (approach along straight lines of varying slope m) or y = x² (approach along a parabola) and check whether the result depends on the choice. For f(x,y) = xy/(x² + y²), the limit along y = 0 gives 0, but along y = x gives x²/(2x²) = 1/2. Because two paths give different values, no limit exists at the origin. The path test can efficiently disprove a limit's existence, but it can never prove existence — checking finitely many paths leaves infinitely many unchecked.
To prove a limit exists, you need the full epsilon-delta definition with the Euclidean distance r = √[(x−a)² + (y−b)²] replacing |x−a|. The strategy is to bound |f(x,y) − L| in terms of r and show the bound goes to zero. Polar coordinates (x = a + r cos θ, y = b + r sin θ) are often the cleanest tool near the origin: r → 0 corresponds to approaching along any path whatsoever, and if the expression in polar form tends to L independently of θ, the limit is established for all paths at once. The squeeze theorem applies in exactly the same way as in one dimension.
Continuity at (a, b) means three things hold simultaneously: f is defined there, the limit exists, and the limit equals f(a, b). Polynomials, exponentials, and trigonometric functions are continuous everywhere they are defined — for these, limit evaluation is just substitution. The interesting cases are rational functions where the denominator vanishes, and piecewise-defined functions where you must check whether the pieces agree at the boundary. Continuity is the foundational hypothesis for everything that follows: partial derivatives assume it, the chain rule requires it, and differentiability (which is stronger than continuity) implies it. Getting comfortable with the multi-path nature of limits is the essential step before any further calculus in higher dimensions.