- 1.
*f*is defined at*P*_{0};- 2.
*f*has a limit at*P*_{0};- 3.
- this limit is equal to
*f*(*P*_{0}).

As a consequence of these definitions and of Prop. 4.7, we have:

- A polynomial in two variables (
*x*,*y*) defines a continuous function on . - A rational function (i.e. the quotient of two polynomial functions) is continuous on its domain.

- 1.
- Let
*f*be a function of two real variables (*x*,*y*), defined on a domain*D*and let*u*and*v*be two functions of the real variable*t*, defined on the same interval*I*. Moreover suppose that*u*and*v*are continuous on*I*and that*f*is continuous on*D*. Then the function is continuous on*I*. - 2.
- Let
*f*be a function of two real variables (*x*,*y*), defined on a domain*D*and let be a function of one real variable, defined on an interval*J*. Moreover suppose that*f*is is continuous on*D*, is continuous on*J*and that for every , we have . Then the function is continuous on*D*.

- The function is continuous on .
- The function
is continuous on the complementary of the
*y*-axis. - The function is continuous on the closed unit disk.