Show that f(x) = |x − 3| is continuous but not differentiable at x = 3.

1. 1 · Write what is given and what is to be found. Given: f(x) = |x − 3|. To show: f is continuous at 3 but not differentiable at 3.
2. 2 · Write the definitions. Continuous at c: LHL = RHL = f(c). Differentiable at c: LHD = RHD.
3. 3 · Check continuity.
   • f(x) = 3 − x for x < 3 and x − 3 for x ≥ 3
   • LHL = 0, RHL = 0 and f(3) = 0
   • So f is continuous at x = 3
4. 4 · Find the LHD. lim (h → 0⁻) [f(3 + h) − f(3)]/h = lim |h|/h = lim (−h)/h = −1.
5. 5 · Find the RHD. lim (h → 0⁺) |h|/h = lim h/h = 1.
6. 6 · Check and box the answer. LHD = −1 ≠ RHD = 1, so f is not differentiable at 3. Hence f is continuous but not differentiable at x = 3.

Scoring tip: Write |h| = −h for h < 0 clearly. That one line carries the method mark.