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Home Partial Differentiation Contionuous Functions of Two or More Variables Theorem 2
 
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Theorem 2

The next theorem shows that most functions we deal with are continuous.

THEOREM 2

(i) If f(x, y) is continuous at (a, b) and g(u) is continuous at f(a, b), then h(x,y) = g(f(x,y)) is continuous at (a, b).

(ii) Sums, differences, products, quotients, and exponents of continuous functions are continuous.

PROOF

(i) If (x, 3,) ≈ (a, b) then f(x, y) ≈ f(a, b), hence g(f(x, y)) ≈ g( f(a, b)), and thus h(x, y) ≈ fr(a, b).

(ii) Let f(x, y) and g(x,y) be continuous at (a, b). As an illustration we show that if f(x,y) > 0 then h(x,y) = f(x,y)g(x,y) is continuous at (a, b). Let (x, y) ≈ (a, b). Then st(h(x, y)) = st(f(x, y)g(x,y)) = st(f(x, y))st(g(x,y)) = f(a, b)g(a,b) = h(a, b).

A function is said to be continuous on a set S of points in the plane if it is continuous at every point in S. Thus the quotient function f(x, y) = x/y is continuous on the set of all (x, y) such that y ≠ 0. The function f(x, y) = xy is continuous on the set of all (x, y) such that x > 0.
Example 2&3
Example 4
Example 5
Example 6

Continuous functions of three or more variables are defined in the natural way, and Theorem 2 holds for such functions.

Example 7

Home Partial Differentiation Contionuous Functions of Two or More Variables Theorem 2

Last Update: 2006-11-05