Consider the function g = |x| in the interval [-1, 1]. In this interval, the function is A. continuous and differentiable B. continuous but not differentiable C. differentiable but not continuous D. neither continuous nor differentiable

GATE 1998 · Engineering Mathematics · Continuity · easy

Answer: g = |x| is continuous but not differentiable on [-1, 1]. Answer: B.

1. Check continuity on [-1, 1]: lim_{x->0^-} |x| = 0, lim_{x->0^+} |x| = 0, f(0) = 0. All equal, so |x| is continuous at x = 0 (and clearly continuous elsewhere). Conclusion: continuous on [-1,1].
2. Check differentiability at x = 0: Left derivative: lim_{h->0^-} (-h)/h = -1. Right derivative: lim_{h->0^+} h/h = +1. Since -1 != +1, f is NOT differentiable at x = 0.