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If f : R ->R is a function defined by f(x) = [ x − 1 ] c o s ( 2 x − 1 2 ) π ,  where[.] denotes the greatest integer function, then f is:

Option 1 -

Discontinuous only at x = 1

Option 2 -

Discontinuous at all integral value values of x except at x = 1

Option 3 -

Continuous for every real x

Option 4 -

Continuous only at x = 1

0 3 Views | Posted a month ago
Asked by Shiksha User

  • 1 Answer

  • A

    Answered by

    alok kumar singh | Contributor-Level 10

    a month ago
    Correct Option - 3


    Detailed Solution:

    f ( x ) = [ x − 1 ] c o s ( 2 x − 2 ) π           

    I f     x = k ,         k ∈ I

    then f(x) = 0 as c o s ( 2 k − 1 2 ) π = 0 , ∀ k ∈ I  

    LHL = L t h → 0 [ k − h − 1 ] c o s ( 2 k − 2 h − 1 2 ) π = L t h → 0 ( k − 2 ) c o s ( 2 k − 1 2 ) π → 0  

    R H L = L t h → 0 [ k + h − 1 ] c o s ( 2 k + 2 h − 1 2 ) π = L t h → 0 ( k − 1 ) c o s ( 2 k − 1 2 ) π → 0        

    ∴ f ( x ) is continuous ∀ x ∈ R  

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Kindly go through the solution 

 

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