alleryConsider $f(x) = |x|$ at $x = 0$. Which statement is correct?
ADifferentiable but not continuous at $x = 0$
BContinuous and differentiable at $x = 0$
CNeither continuous nor differentiable at $x = 0$
DContinuous but not differentiable at $x = 0$
Answer & Solution
Correct answer: D. Continuous but not differentiable at $x = 0$
**Continuity at $0$.** $\lim_{x\to 0} |x| = 0 = f(0)$, so $f$ is continuous.
**Differentiability at $0$.** Compute the one-sided derivatives:
- $f'(0^+) = \lim_{h\to 0^+} \dfrac{|h| - 0}{h} = \lim_{h\to 0^+} \dfrac{h}{h} = +1$.
- $f'(0^-) = \lim_{h\to 0^-} \dfrac{|h| - 0}{h} = \lim_{h\to 0^-} \dfrac{-h}{h} = -1$.
The left and right derivatives disagree, so $f'(0)$ does not exist
→ $f$ is **not** differentiable at $0$.
**Takeaway.** Differentiability implies continuity, but the converse fails — $|x|$ is the canonical counterexample.
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