Question

If f is continuous at x_0 and g is continuous at f(x_0), then the composite function g o f is

A. Continuous at x_0.
B. Discontinuous at x_0.
C. Undefined at x_0.
D. Discrete at x_0.

Answer 1

The composite function g o f is continuous at x_0 when f is continuous at x_0 and g is continuous at f(x_0).

Step-by-step explanation:

If f is continuous at x_0 and g is continuous at f(x_0), then the composite function g o f is Continuous at x_0.

A function is said to be continuous at a point if the function is defined at that point, the limit of the function as it approaches the point exists, and the value of the function at that point is equal to the limit. Since f is continuous at x_0, by definition, f(x) approaches f(x_0) as x approaches x_0, and f(x) is defined at x_0. Similarly, since g is continuous at f(x_0), as y approaches f(x_0), g(y) approaches g(f(x_0)). Therefore, as x approaches x_0, g(f(x)) approaches g(f(x_0)), implying that g o f is continuous at x_0.