Question

For the function f(x,y) below, determine whether there is a value for c making the function continuous everywhere. If so, find it.

f(x,y)= { c+y x ≤ 3,
{ 3-y x >3.
c = ____

Answer 1

To determine if the function f(x,y) is continuous everywhere, we need to check if the function is continuous at x=3. To be continuous at x=3, the value of the function from the left side, which is c+y, must be equal to the value of the function from the right side, which is 3-y. Therefore, c = (3-2y) makes the function continuous at x=3.

Step-by-step explanation:

To determine if the function f(x,y) is continuous everywhere, we need to check if the function is continuous at x=3, since that is the only point where the function is defined differently for x≤3 and x>3.

To be continuous at x=3, the value of the function from the left side, which is c+y, must be equal to the value of the function from the right side, which is 3-y. Therefore, we can set c+y = 3-y and solve for c.

Adding y to both sides and rearranging the equation, we get c = (3-2y). So, there is a value for c that makes the function continuous at x=3, and the value of c is (3-2y).