Question

Let f and g be real-valued functions that are continuous at x_0 in R. Then

1) f+g is ______ at x_0.
2) fg is ____ at x_0.
3) f/g is ______ at x_0 if g(x_0) does not equal ___ .

Answer 1

The functions f+g and fg are continuous at x_0. The function f/g is also continuous at x_0, provided that g(x_0) is not equal to 0.

Step-by-step explanation:

If f and g are real-valued functions that are continuous at x_0 in R, then:

1. f+g is continuous at x_0.
2. fg is continuous at x_0.
3. f/g is continuous at x_0 if g(x_0) does not equal 0.

The reason these statements are true is that the sum, product, and quotient (where the denominator is not zero) of continuous functions are also continuous. These operations can be performed without causing any breaks or jumps in the resulting function at points where both original functions are continuous.