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  • The function f(x)=[x] cos [(2x-1)/2] pi ,( where [.] denote the greatest integer function ) is discontinuous at

The function f(x)=[x] cos [(2x-1)/2] pi ,( where [.] denote the greatest integer function ) is discontinuous at

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Solution:   Given , 

               f(x)=[x]cos [frac(2x-1)2]pi

    Case (1)     When  xin n, then  f(x)=xcos [frac(2x-1)2]pi

                            f(n)=cos (n-1)pi\ \ lim_x o nf(x)=ncos (n-1)pi\ \ lim_x o nf(x)=(n-1)cos (n-1)pi

herefore      Limit exist , if  cos (n-1)pi=0  which is not possible 

herefore      f(x)  is discontinuous at all xin I.

Case (2)    When x is not integer 

Let              frac2x-12=m,m is integers then

                   x=frac2m+12=m+frac12\\ Rightarrow lim_x o m+frac12^-f(x)=mcos (m-1)pi\ \ Rightarrow lim_x o m+ frac12^+f(x)=mcos m pi

herefore     Limit exist only when   m=0  i.e  n=frac12

Hence , discontinuous at   x=fracn2,nin I-1

Posted by

Deependra Verma

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