2.1 Functions of two variables

Let $(x,y)$ be the co-ordinates of a point in the plane. Suppose that we have a function $f:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ that assigns to each pair $(x,y)$ a value $z=f(x,y)$. Then $f$ is a function of the two variables $x$ and $y$, which can vary independently.

Illustration. Let $(x,y)$ be the co-ordinates defining point on a map, so that the $y$-axis points northwards and the $x$-axis points eastwards. We let $f(x,y)$ be the height of the land at $(x,y)$.

In the usual 3-dimensional co-ordinate axes $0xyz$, the domain of $f$ is a subset of the $(x,y)$-plane, and the codomain is in the $z$-axis. The graph of $f$ is $z=f(x,y)$, which gives a surface. We say that $f$ is continuous if the surface is smooth so that $f(x,y)\rightarrow f(a,b)$ as $(x,y)\rightarrow(a,b)$ in any way.