Limits and continuity

For functions of one variable, the (rough) definition of a limit was:

${\displaystyle \lim_{x \to a} f(x) = L}$ if $f(x)$ is close to $L$ whenever $x$ is close to (but not equal to) $a$.

We made that precise by saying exactly what `close to' means. We used the letter $\epsilon$ for how close $f(x)$ has to get to $L$, and $\delta$ for how close $x$ is to $a$:

Precise definition of limit: ${\displaystyle \lim_{x \to a} f(x) = L}$ if, for every number $\epsilon > 0$ there exists a number $\delta >0$ such that $|f(x) -L| < \epsilon$ whenever $0<|x-a|<\delta$.

The rough and precise definitions of limits of functions of two (or more) variables work the same way:

(Rough definition) ${\displaystyle \lim_{(x,y) \to (a,b)} f(x,y) = L}$ if $f(x,y)$ is close to $L$ whenever the point $(x,y)$ is close to (but not equal to) $(a,b)$. (Precise definition) ${\displaystyle \lim_{(x,y) \to (a,b)} f(x,y) = L}$ if, for every number $\epsilon > 0$ there exists a number $\delta >0$ such that $|f(x) -L| < \epsilon$ whenever $0<\|\langle x-a, y-b \rangle\|<\delta$.

The only difference is what we mean by `close to $(a,b)$'. We mean the distance in the plane: $$\|\langle x-a, y-b \rangle\| = \sqrt{(x-a)^2 + (y-b)^2}.$$ This is small whenever $x$ is close to $a$ and $y$ is close to $b$. Just one of them being close isn't good enough.

Example 1: Is $\displaystyle{\lim_{(x,y) \to (2,3)} xy^2 = 18}$?

Solution: Yes, since a number close to 2 times the square of a number close to 3 is close to $2(3)^2=18$.

Example 2: If $$f(x,y) = \frac{x^2-y^2}{x^2+y^2},$$ then what is $\displaystyle{\lim_{(x,y) \to (0,0)} f(x,y)}$?

Solution: This limit does not exist. You can find points arbitrarily close to the origin with $f(x,y)=1$, $f(x,y)=-1$ or anything in between. For instance, $f(0.00001,0)=1$, $f(0,0.00001)=-1$, and $f(0.00001, 0.00001)=0$. (When written in polar coordinates, $f(x,y)=\cos^2(\theta)-\sin^2(\theta) = \cos(2\theta)$.)

Just as with one variable, we say a function is continuous if it equals its limit:

A function $f(x,y)$ is continuous at the point $(a,b)$ if $\displaystyle{\lim_{(x,y) \to(a,b)}f(x,y) = f(a,b)}$. A function is continuous on a domain $D$ if is is continuous at every point of $D$.

Most of the rules for continuous functions carry over unchanged from single variable calculus. For instance,

Polynomials in $x$ and $y$ are continuous. Compositions of continuous functions are continuous. (E.g. $\sin(x^2+y^2)$ is continuous since $x^2+y^2$ is a continuous function of two variables and $\sin(t)$ is a continuous function of its input.) Sums and products of continuous functions are continuous. Ratios of continuous functions are continuous, except where the denominator goes to zero.