From two doors to infinitely many
In Volume I a limit of a single-variable function had exactly two doors. To say the limit of f(x) as x approaches a equals L, you only had to check that the left-hand and right-hand approaches agreed: come in from below, come in from above, and if both land on L, you are done. The whole real line offers a point only two sides. That tidy bookkeeping is why one-sided limits settle almost everything in first-year calculus.
Now lift the picture into the plane. A function f(x, y) lives over a flat sheet, and a point like (0, 0) is no longer pinned between two neighbors — it sits at the center of a whole disk. You can walk toward it straight along the x-axis, or along the y-axis, or down any slanted line, or spiralling in along a curve, or wobbling in along a parabola. There are not two ways to approach; there are infinitely many. The [[limit-of-a-function-of-several-variables|limit of a function of several variables]] demands that every single one of these approaches deliver the same value L.
Stated carefully, the limit of f(x, y) as (x, y) approaches (a, b) is L if you can force f to stay within any tolerance of L just by keeping the point close enough to (a, b). The natural ruler of closeness is the distance sqrt((x - a)^2 + (y - b)^2). This is the same epsilon-delta promise as before — for every epsilon there is a delta — except that 'close enough' now means inside a small disk, not inside a small interval. The promise must hold no matter how the point enters that disk.
The surprise: lines agree, the limit still fails
Here is the trap that catches everyone once. Consider f(x, y) = xy / (x^2 + y^2) near the origin. Walk in along the x-axis (set y = 0) and the function is 0 / x^2 = 0 the whole way — the limit along that road is 0. Walk in along the y-axis (set x = 0) and again you get 0. Try any straight line y = mx and you might expect 0 once more. So does the limit equal 0? It is tempting to say yes — and it is wrong.
Substitute y = mx into the formula. Numerator xy = m x^2; denominator x^2 + y^2 = x^2(1 + m^2). The x^2 cancels and you are left with m / (1 + m^2) — a constant that depends on the slope. Approach along y = x (slope m = 1) and f sits at 1/2 the entire way. Approach along the x-axis (slope 0) and f sits at 0. Two different straight roads, two different destinations. Because the answer changes with direction, the two-variable limit at the origin simply does not exist. This is the heart of [[path-dependence-of-limits|path dependence]].
Proving a limit really does exist
If trying paths can only ever disprove a limit, how do you prove one exists? You bound the whole function at once, with no mention of direction. The cleanest tool is the multivariable squeeze theorem: trap |f(x, y) - L| between 0 and some quantity that you can prove shrinks to 0 as the distance r = sqrt(x^2 + y^2) shrinks to 0. If the upper bound depends only on r and goes to 0, then every path is squeezed to L simultaneously, because every path makes r go to 0.
Watch it work on h(x, y) = x^2 y / (x^2 + y^2), which looks dangerously like the function that failed. Because x^2 is never larger than x^2 + y^2, the fraction x^2 / (x^2 + y^2) is at most 1. So |h| = |y| times that fraction is at most |y|. And |y| is at most r, the distance to the origin. Therefore 0 <= |h(x, y)| <= r, and as r goes to 0 the squeeze forces h to 0. No direction was ever mentioned, so the limit is 0 along every path. This function is genuinely continuous at the origin; its near-twin from the last section was not.
A second trick is to switch to polar coordinates: write x = r cos(theta), y = r sin(theta). Now (x, y) approaching the origin is exactly r approaching 0, with theta free to be anything — theta carries the direction information. If, after substituting, the expression collapses to something bounded by a power of r with the theta-dependence trapped inside bounded factors like cos(theta) and sin(theta), the limit exists and equals whatever the r-to-0 part gives. If instead theta refuses to drop out — if the answer still depends on theta after r cancels — you have just exposed path dependence, and the limit does not exist.
What continuity means now
With limits understood, [[continuity-of-several-variables|continuity of a function of several variables]] reads exactly as it did in one variable, just with the new kind of limit plugged in. The function f is continuous at (a, b) when three things line up: f is actually defined at (a, b); the limit of f as (x, y) approaches (a, b) exists; and that limit equals the value f(a, b). No surprise jumps, no holes you could fall through — the surface meets its own value smoothly from every direction at once.
The good news is that the menagerie of failures lives almost entirely at a handful of special points where a denominator vanishes. Everywhere else the ordinary algebra of continuity carries over untouched: sums, products, quotients (away from zero denominators), and compositions of continuous functions are continuous. Polynomials in x and y are continuous on the whole plane; sin(x + y), e^{xy}, and sqrt(1 + x^2 + y^2) are continuous wherever they are defined. You only have to slow down and think hard at the isolated trouble spots — typically where a fraction reads 0/0.
f continuous at (a,b) requires all three: 1. f(a,b) is defined 2. lim (x,y)->(a,b) f(x,y) exists <- the hard one: EVERY path 3. the limit = f(a,b) repair a 0/0 hole? define f(a,b) := the limit, IF the limit exists
Why this matters for everything ahead
This is not a curiosity — it is the bedrock for the whole rung. Recall from Volume I that a partial derivative takes a limit while holding the other variables frozen, so it only ever feels two of the infinitely many directions: along x and along y. That is exactly why partial derivatives can exist at a point while the function is not even continuous there. The pathological xy/(x^2 + y^2) has both partials equal to 0 at the origin, yet it has no limit there at all. Partial derivatives alone are a weak, axis-bound probe.
That gap is precisely what the next guide repairs. True [[differentiability-several-variables|differentiability in several variables]] is a far stronger, direction-blind condition: it asks that one single flat plane approximate the surface well from every direction at once, with the error shrinking faster than the distance r. Continuity is the floor that strong condition stands on — a function that is differentiable at a point is automatically continuous there, never the wrong way around. Get the 'every path' instinct into your bones here, and the total derivative, the Jacobian, and the full chain rule in the guides ahead will feel like natural next steps rather than new mysteries.