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 h(x,y) is the product of two continuous functions f(x) and g(y), both of which are continuous on their respective intervals [a,b] and [c,d]. As such, h(x,y) is also continuous on the rectangular domain R = [a,b] x [c,d] by the product rule for continuous functions. This means that for any point (x,y) in R, we can make h(x,y) arbitrarily close to a fixed value L by choosing (x,y) sufficiently close to (a,b).

Determine Whether the Statement Is True or False