The function f(x)=[x]n−xn,n≥2 (where[x] is the greatest integer less than or equal to x), is discontinuousat all points of

The function f(x)=[x]nxn,n2 (where
[x] is the greatest integer less than or equal to x), is discontinuous
at all points of

  1. A

    I

  2. B

    I~{0,1}

  3. C

    I~{0}

  4. D

    I~{1}

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    Solution:

    f(x)=0 for xI

    If 0<x<1, then 0<xn<1, so 

    =0 and xn=0 f(x)=0 for 0<x<1

    If 1<x<21/n then 1<xn<2[x]=1

    Thus f(x)=[x]nxn=0 if 1<x<21/n

    So f(x)=0 if 0x<21/n

    Therefore, f is continuous at x = 1
    To the left of any integral value m1 but close to m, f(x)

    0 but to the right of m and close to m, f(x) = 0. Hence f is

    discontinuous for all mI~{1}

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