If f : R → R is a function defined by f(x) = [x – 1] cos ((2x – 1)/2)π, where [.] denotes the greatest integer function, then f is:

  1. discontinuous only at x = 1
  2. discontinuous at all integral values of x except at x = 1
  3. continuous only at x = 1
  4. continuous for every real x

Answer (Detailed Solution Below)

Option 4 : continuous for every real x

Detailed Solution

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Concept:

A function f(x) is continuous at x = a if limxa f(x)=limxa+ f(x)=limxa f(x)

Calculation

Doubtful points are x = n, n∈I

LHL = limxn[x1]cos(2x12)π=(n2)cos(2n12)π=0

RHL = limxn+[x1]cos(2x12)π=(n1)cos(2n12)π=0

⇒  f(x) = 0 ∀ x ∈ R

∴ f is continuous for every real x.

The correct answer is Option 4.

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