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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       

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 $t \mapsto f(u(t), v(t))$ is continuous on I.
2.
Let f be a function of two real variables (x,y), defined on a domain D and let $\phi$ be a function of one real variable, defined on an interval J. Moreover suppose that f is is continuous on D, $\phi$ is continuous on J and that for every $(x,y) \in D$, we have $f(x,y) \in J$. Then the function $(x,y) \mapsto \phi ( f(x,y))$ is continuous on D.

Example 4.16       


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