Limit and Continuity of Functions of Two Variables

Definition 8.6 (Limit of a Function of Two Variables). Let $A=\{(x,y)\mid a<x<b,\,c<y<d\}\subset\mathbb R^2$ and $F:A\to\mathbb R$. $F$ has a limit $L$ at $(u,v)$ if: for every neighbourhood $(L-\varepsilon,L+\varepsilon)$, $\varepsilon>0$, of $L$, there exists a $\delta$-neighbourhood $B_\delta((u,v))\subset A$ of $(u,v)$ such that

We write $\displaystyle\lim_{(x,y)\to(u,v)}F(x,y)=L$ if such a limit exists. All the standard limit theorems (limits of sums, differences, products, quotients — where the denominator's limit is nonzero — and composition with a continuous function) that hold for one-variable limits hold, unchanged in form, for functions of several variables.

Definition 8.7 (Continuity). $F$ is continuous at $(u,v)$ if: (1) $F$ is defined at $(u,v)$; (2) $\displaystyle\lim_{(x,y)\to(u,v)}F(x,y)$ exists; and (3) that limit equals $F(u,v)$ — the same three-part test as one variable, carried over verbatim.

Worked example — a limit that fails to exist. Consider $f(x,y)=\dfrac{xy}{x^2+y^2}$ for $(x,y)\ne(0,0)$, $f(0,0)=0$. Along any straight line $y=mx$ through the origin,

a value that genuinely depends on the slope $m$ of the approach line — different lines through the origin give different limiting values (e.g. $m=0$ gives $0$, $m=1$ gives $\tfrac12$). Since the limit is not the same along every path, $\displaystyle\lim_{(x,y)\to(0,0)}f(x,y)$ does not exist, and $f$ is consequently not continuous at $(0,0)$.

Worked example — establishing continuity via a bound. Let $g(x,y)=\dfrac{x^2y}{x^2+y^2}$ for $(x,y)\ne(0,0)$, $g(0,0)=0$; this is continuous everywhere, including at the origin. Away from the origin it is a quotient of continuous functions with nonvanishing denominator, hence continuous there directly. At $(0,0)$:

using $\dfrac{x^2}{x^2+y^2}\le1$. Since $(x,y)\to(0,0)$ forces $|y|\to0$, the squeeze gives $\displaystyle\lim_{(x,y)\to(0,0)}g(x,y)=0=g(0,0)$, so $g$ is continuous at $(0,0)$ too — hence continuous on all of $\mathbb R^2$.

Working method summary for a two-variable limit/continuity problem:

1. Direct substitution whenever the expression is built from continuous pieces (polynomials, $\sin$, $\cos$, $e^{(\cdot)}$, $\log$, ...) combined algebraically or by composition, with any denominator nonzero at the target point. …