Question

Suppose f(x, y) is continuous and nonnegative over a region r in the plane with a defined area A(r).

a) The area A(r) is irrelevant to the continuity of f(x, y).

b) There is not enough information to draw conclusions about f(x, y).

c) f(x, y) must be a constant function.

d) The integral of f(x, y) over r is nonnegative.

Answer 1

The area A(r) is irrelevant to the continuity of f(x, y).

Step-by-step explanation:

The area A(r) is irrelevant to the continuity of f(x, y).

The continuity of a function depends on the behavior of the function as it approaches a particular point. It is determined by the limit of the function as x and y approach that point.

The area under the function does not affect its continuity. Even if the function has a large or small area, it can still be continuous.

For example, consider the function f(x, y) = x^2 + y^2. This function is continuous everywhere, but the area under it can vary depending on the region.

Therefore, the correct answer is a) The area A(r) is irrelevant to the continuity of f(x, y).