Question

The function $f(x) = |x|$ is:

• A. continuous and differentiable at $x=0$
• B. continuous but not differentiable at $x=0$
• C. neither continuous nor differentiable at $x=0$
• D. differentiable but not continuous at $x=0$

Hint:

A "sharp corner" or "v-shape" on a graph indicates a point where the function is not differentiable.

Answer 1

The Correct Option is B

Step 1: Concept
Examine the limit and the derivative of the absolute value function at the origin.
Step 2: Analysis
$\lim_{x \to 0} |x| = 0 = f(0)$, so it is continuous. Left-hand derivative is $-1$ and right-hand derivative is $+1$.
Step 3: Conclusion
Since the left and right derivatives are not equal, the function is not differentiable at $x=0$.
Final Answer: (B)