continuity of several variables
A surface z = f(x, y) is continuous at a point when it has no holes, no cliffs, and no sudden tears there — you could lay your hand flat on it without the surface jumping away under your fingers. Continuity of several variables is exactly this no-surprises property made precise: small steps in the input never cause a sudden jump in the output.
Formally, f is continuous at (a, b) when three things all hold and agree: f(a, b) is defined, the limit of f(x, y) as (x, y) -> (a, b) exists, and that limit equals f(a, b). Because the limit here is the genuine two-dimensional limit, it must be the same along every path in. So a multivariable function can fail continuity in a way impossible in one dimension: it can be perfectly defined at the point and yet have no limit there, because different directions of approach disagree. The function is continuous on a region when it is continuous at every point of that region.
Continuity is the entry ticket to every theorem that makes calculus usable. A continuous function on a closed bounded region attains a maximum and a minimum (Weierstrass), which is what lets optimization look for them; it can be integrated; and continuity is necessary (though not sufficient) for differentiability. Sums, products, quotients (away from zeros), and compositions of continuous functions are continuous, so in practice you rarely check the definition directly — you build continuous functions out of continuous pieces and only worry at the special points where a denominator vanishes or a formula is patched.
Define f(x, y) = xy / (x^2 + y^2) for (x, y) not (0, 0) and f(0, 0) = 0. The value at the origin is defined, but the limit there does not exist, so f is discontinuous at (0, 0) — yet continuous everywhere else, since it is a quotient of polynomials with nonzero denominator off the origin.
Patching a value at a bad point cannot create continuity if the underlying limit fails to exist.
Continuity in each variable separately (fix y, vary x; then fix x, vary y) is much weaker than joint continuity — a function can be continuous along every axis-parallel line yet discontinuous at the point. Always check the genuine two-variable limit.