Question

Let $f(x) = |x|$, $x \in \mathbb{R}$. Then, which of the following statements is incorrect?

• A. $f$ has a minimum value at $x = 0$
• B. $f$ has no maximum value in $\mathbb{R}$
• C. $f$ is continuous at $x = 0$
• D. $f$ is differentiable at $x = 0$

Hint:

The absolute value function $f(x) = |x|$ is not differentiable at $x = 0$, although it is continuous.

Answer 1

The Correct Option is D

The function $f(x) = |x|$ exhibits continuity at $x = 0$. This is due to the limit of $f(x)$ as $x$ approaches 0 from either direction equaling the function's value at $x = 0$. Nevertheless, $f(x)$ lacks differentiability at $x = 0$ because the left-hand and right-hand derivatives do not coincide at this point. Consequently, the derivative is undefined at $x = 0$. Therefore, the assertion that $f$ is differentiable at $x = 0$ is inaccurate.