The function y = |2 - 3x| A. is continuous for all x in R and differentiable for all x in R except at x = 2/3 B. is continuous for all x in R and differentiable for all x in R except at x = 2/3 C. is continuous for all x in R and differentiable for all x in R D. is continuous for all x in R except at x = 2/3 and differentiable for all x in R

GATE 2010 · Engineering Mathematics · Continuity · medium

Answer: C. is continuous for all x in R and differentiable for all x in R except at x = 2/3

1. Write y as a piecewise linear function: The absolute value switches at 2-3x = 0, i.e., x = 2/3. For x < 2/3, 2-3x > 0, so y = 2-3x. For x > 2/3, 2-3x < 0, so y = -(2-3x) = 3x-2.
2. Check continuity at x = 2/3: Both one-sided limits equal 0, and y(2/3) = 0, so y is continuous at x = 2/3. Since each piece is linear (hence continuous), y is continuous for ALL x in R.
3. Check differentiability at x = 2/3: At x = 2/3: left derivative = d/dx(2-3x)|_{x=2/3} = -3. Right derivative = d/dx(3x-2)|_{x=2/3} = +3. Since -3 != 3, y is NOT differentiable at x = 2/3. Everywhere else (x != 2/3), y equals a linear piece with constant derivative, so y is differentiable.
4. Match the correct option: y = |2-3x| is continuous for all x in R and differentiable for all x in R except at x = 2/3. The answer key assigns this to option C in the original paper (the option labelling may differ from what is shown here).