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Find all the points of discontinuity of ...

Find all the points of discontinuity of the greatest integer function defined by `f(x) = [x]`, where [x] denotes the greatest integer less than or equal to x.

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To find all the points of discontinuity of the greatest integer function defined by \( f(x) = [x] \), where \([x]\) denotes the greatest integer less than or equal to \( x \), we will analyze the function step by step. ### Step 1: Understanding the Greatest Integer Function The greatest integer function, \([x]\), returns the largest integer that is less than or equal to \( x \). For example: - \([2.3] = 2\) - \([3] = 3\) - \([-1.5] = -2\) ...
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Knowledge Check

  • The function of f:R to R , defined by f(x)=[x] , where [x] denotes the greatest integer less than or equal to x, is

    A
    one-one
    B
    onto
    C
    one-one and onto
    D
    neither one-one nor onto

  • If f(x)=|x-1|-[x] , where [x] is the greatest integer less than or equal to x, then

    A
    `underset(xto0)lim[f(x)]=0`
    B
    `underset(xto0)lim[f(x)]=1`
    C
    `underset(xto0)lim[(f(x))/(x)]` does not exist
    D
    `underset(xto0)lim[(f(x))/(x)]` exists

  • The domain of the function f(x)=cos^(-1)[secx] , where [x] denotes the greatest integer less than or equal to x, is

    A
    `{x:x=(2n+1) pi, n in Z} cup { x : 2 m pi le x lt 2m pi+(pi)/(3), m in Z}`
    B
    `{x:x=2n pi , n in Z} cup { x: 2 m pi lt(x =2n pi) , n in Z} cup { x : 2 m pi lt x lt 2m pi +(pi)/(3), m in Z }`
    C
    `{x:(2n+1) pi, n in Z} cup { x : 2m pi lt x lt 2 m pi +(pi)/(3), m in Z}`
    D
    none of these

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