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MHT CET202611 April 2026Evening ShiftMathematicsContinuity and DifferentiabilityActual

Let the function f be defined by f(x) = x - |x| x , for x 0 and f(0) = 2 then f is

Options

  1. Acontinuous nowhere
  2. Bcontinuous for all x except at x = 0
  3. Ccontinuous everywhere
  4. Dcontinuous for all x except at x = 1

Correct answer

B. continuous for all x except at x = 0

Step-by-step solution

Given f(x) = x - |x| x for x 0 and f(0) = 2 For x > 0 , |x| = x , so f(x) = x - x x = 0 For x Left-hand limit at x = 0 is _ x 0^- f(x) = 2 Right-hand limit at x = 0 is _ x 0^+ f(x) = 0 Since _ x 0^- f(x) _ x 0^+ f(x) , the function is discontinuous at x = 0 . For x 0 , the function is constant ( 0 for x > 0 and 2 for x Thus, f(x) is continuous for all x except at x = 0 . Answer: continuous for all x except at x = 0

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