Next: Partial derivatives; the differential Up: Multivariable functions. Previous: Limits.

# Continuous functions.

Definition 4.12   The function f is continuous at the point P0=(x0,y0) if the following properties are fulfilled:
1.
f is defined at P0;
2.
f has a limit at P0;
3.
this limit is equal to f(P0).

Definition 4.13   Let f be a function defined on an open domain D. The function f is continuous on D if it is continuous at every point of D.

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

Corollary 4.14

• 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.

Proposition 4.15

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.

Example 4.16

• 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.

Next: Partial derivatives; the differential Up: Multivariable functions. Previous: Limits.
Noah Dana-Picard
2001-05-30