Consider $f(x) = |x|$ at $x = 0$. Which statement is correct?

**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.