Determine whether the statement is true or false.If it is true, explain why it is true.If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where
A)From the definition follows , so the statement is true.
B)From the definition follows , so the statement is false.
C)From the definition follows , so the statement is false.

Question

Determine whether the statement is true or false.If it is true, explain why it is true.If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then  where 
A)From the definition follows , so the statement is true. 
B)From the definition follows , so the statement is false. 
C)From the definition follows , so the statement is false.

Quizplus · Accepted Answer

The statement is true because it follows from the definition of continuity and the product rule for continuous functions. Since f is continuous on [a, b], it can be shown that g(y) approaches g(c) as x approaches a, and g(d) as x approaches b. Therefore, h(x, y) approaches f(a)g(c) as (x, y) approaches (a, c), and f(b)g(d) as (x, y) approaches (b, d). By the product rule for continuous functions, h(x, y) is continuous on [a, b] x [c, d].

Determine Whether the Statement Is True or False