Question

Let f be a real-valued function with Dom(f) ⊂ R. If f is continuous at x0 in Dom(f), then |f| and kf, where k exists in R, are _____ at x0.

A. Continuous.
B. Discontinuous.
C. Bounded.
D. Unbounded.

Answer 1

If a real-valued function f is continuous at x0 in Dom(f), then both the absolute value of f (|f|) and a scalar multiple of f (kf, where k exists in R) are continuous at x0.

Step-by-step explanation:

If a real-valued function f is continuous at x0 in Dom(f), then both the absolute value of f (|f|) and a scalar multiple of f (kf, where k exists in R) are continuous at x0. If f is a real-valued function with Dom(f) ⊂ R, and f is continuous at x0 in Dom(f), then |f| and kf, where k exists in R, are also continuous at x0. This is because the absolute value function and the scalar multiplication (by a real number) are continuous operations. If f is continuous at a point, taking the absolute value of f will also be continuous, and multiplying f by any real number k will not affect its continuity. f remains bounded within its domain, and these operations do not introduce any discontinuities.