Consider the following two statements about the function f(x) = |x|: P. f(x) is continuous for all real values of x. Q. f(x) is differentiable for all real values of x. Which of the following is TRUE? A. P is true and Q is false. B. P is false and Q is true. C. Both P and Q are true. D. Both P and Q are false.

GATE 2007 · Engineering Mathematics · Continuity · easy

Answer: P is true, Q is false. Answer: A.

1. Verify P — continuity at x = 0: lim_{x->0^-} (-x) = 0 and lim_{x->0^+} x = 0 and f(0) = 0. All three agree. Away from 0, each piece is a linear function (continuous). So P is TRUE.
2. Falsify Q — differentiability at x = 0: Left derivative at 0: d/dx(-x)|_{x=0^-} = -1. Right derivative at 0: d/dx(x)|_{x=0^+} = +1. Since -1 != +1, f is not differentiable at x = 0. So Q is FALSE.