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JEE Main202124 Feb 2021Morning ShiftMathematicsContinuity and DifferentiabilityActual

If f : R → R is a function defined by f x = x - 1 cos 2 x - 1 2 π , where · denotes the greatest integer function, then f is:

Options

  1. Adiscontinuous only at x = 1
  2. Bdiscontinuous at all integral values of x except at x = 1
  3. Ccontinuous only at x = 1
  4. Dcontinuous for every real x

Correct answer

D. continuous for every real x

Step-by-step solution

Doubtful points are x = n ,   n ∈ I L . H . L   = lim x → n - x - 1 cos 2 x - 1 2 π = n - 2 cos 2 n - 1 2 π = 0 R . H . L   = lim x → n + x - 1 cos 2 x - 1 2 π = n - 1 cos 2 n - 1 2 π = 0 f n = 0 Hence continuous.

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