Question

If $|f(x)|$ is continuous at $x = a$, then $f(x)$

• A. is continuous at $x = a$
• B. is continuous at $x = -a$
• C. is continuous at $x = \sqrt{a}$
• D. need not be continuous at $x = a$

Hint:

$|f(x)|$ continuous $\not\Rightarrow$ $f(x)$ continuous. But if $f(x)$ is continuous, then $|f(x)|$ is always continuous.

Answer 1

The Correct Option is D

Step 1: Conceptual Understanding:
Continuity of a composed function does not necessarily imply continuity of the inner function.
Step 2: Explanation in Detail:
Counter-example: $f(x) = 1$ for $x \ge 0$ and $f(x) = -1$ for $x<0$.
Then $|f(x)| = 1$ is continuous everywhere, but $f(x)$ is discontinuous at $x = 0$.
Step 3: Therefore, Stating the Final Answer
Continuity of $|f(x)|$ does not guarantee continuity of $f(x)$.