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Question

Let [t] denote the greatest integer t. The number of points where the function f(x)=[x]|x21|+sin(π[x]+3)[x+1],x(2,2) is not continuous is

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Solution

f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪2|x21|+1x(2,1)|x21|+1x[1,0)sin(π3)1x[0,1)|x21|+122x[1,2)

limx1f(x)=1 and limx1+f(x)=1
Hence continuous at x=1

limx0f(x)=0 and limx0+f(x)=321
discontinuous

limx1f(x)=321 and limx1+f(x)=122
discontinuous

Hence, 2 points of discontinuity.

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