Question

Which of the following statements is false?

• A. If $f(x)$ is continuous at $x=a,$ then $|f(x)|$ is also continuous at $x=a$
• B. If $f(x)$ is continuous at $x=a,$ then $f^{-1}(x)$ is also continuous at $x=a$
• C. If $|f(x)|$ is continuous at $x=a$, then $f(x)$ is also continuous at $x=a$ of these
• D. None of these

Answer 1

Answer: (C)

Statements (a) and (b) are correct while statement (c) is
incorrect as function $f(x) = \begin{cases} 1, & \text{$x<\,0$ } \\[2ex] -1 & \text{ $x \geq 0$ } \end{cases}$
is discontinuous at x = 0, but $|f(x)|=1$ for $x \in R$ which is continuous at $x = 0$