Let f be a function of two variables whose domain D includes points arbitrarily close to (a,b). Then we say that the limit of f(x,y) as (x,y) approaches (a,b) is L, and we write
if, for every number
there is a corresponding number
such that
whenever 
and
In plain English, what this says is that when we approach (a,b) from any direction, we find ourselves getting closer and closer to L.
A function f of two variables is called continuous at (a,b) if
We say f is continuous on D if f is continuous at every point (a,b) in D.
A polynomial function of two variables is a sum of terms of the form
, where c is a constant and m and n are non-negative integers. A
rational function is a ratio of polynomials.
All polynomials are continuous on 
. All rational functions are
continuous on their domains. More generally, sums and products of continuous
functions are continuous on their domains.
If we approach (a,b) along two different curves, and get two different limits, then the limit
fails to exist.
We see the generalization, from approaching along a 1-dimensional ball (from left and right) in the univariate case, to approaching along a 2-dimensional ball (a circle) in the bi-variate case; in generally, we approach a point along an n-dimensional ball.
The results in this section are a simple generalization of the results from the univariate case. The idea of limit is still the same: the primary difference is that, whereas in the univariate case we were only concerned about limits from the left and from the right, now we are concerned with limits from any direction (and that's 360 degrees)!
It's nice when we can invoke continuity of functions, which works exactly as it did in days past (sum of continuous function is continuous on the mutual domain, etc.). Rational functions are continuous on their domains.
Contour plots can show us a lot about discontinuities, as we can see in the following three images (generated using public domain software gnuplot and the WAVES software:
In the first function we can see gnuplot's valiant efforts to pave over the
discontinuity. It really does look like a cliff in this case:
The next two functions are rational functions, which can only fail to be
continuous at points not on their domains:
As is true in the case of topographic maps, when level curves collide, there's a sharp drop off (a ``cliff''): cliffs mean that limits don't exist, as if you approach from certain directions you hit a ``wall'', rather than making a smooth transition.