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Continuity of a function with [x] or |x| at a point or on a set

This type of question, asked 5 times: 14 May 2024 S1, 20 Apr 2025 S2, 26 Apr 2025 S2, 11 Apr 2026 S2, 20 Apr 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    14 May 2024 S1

    119
    The function f(x)=[x]⋅cos⁡(2x−12π)f(x)=[x]\cdot\cos\left(\dfrac{2x-1}{2}\pi\right), where [⋅][\cdot] denotes the greatest integer function, is discontinuous at

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  2. Question 2

    20 Apr 2025 S2

    137
    f(x)={[x2]−[−x2],x≠3k,x=3f(x) =\left\{ \begin{matrix} \left\lbrack x^{2} \right\rbrack-\left\lbrack -x^{2} \right\rbrack, & x\neq 3 \\ k & ,x= 3 \end{matrix} \right. is continuous at x=3x= 3, then k=k = where [⋅]\lbrack \cdot \rbrack is greatest integer function

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  3. Question 3

    26 Apr 2025 S2

    102
    Let f(x)={x4−5x2+4∣(x−1)(x−2)∣,x≠1,26,x=112,x=2f(x) =\left\{ \begin{matrix} \frac{x^{4}- 5x^{2}+ 4}{|(x- 1)(x- 2)|} & ,x\neq 1,2 \\ 6 & ,x= 1 \\ 12 & ,x= 2 \end{matrix} \right..Then f(x)f(x) is continuous on the set

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  4. Question 4

    11 Apr 2026 S2

    69
    Let the function f be defined by f(x)=x−∣x∣xf(x) = \dfrac{x - |x|}{x}, for x≠0x \neq 0 and f(0)=2f(0) = 2 then f is

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  5. Question 5

    20 Apr 2026 S1

    70
    Let [x][x] denotes the greatest integer less than or equal to x and f(x)=[tan⁡2x]f(x) = [\tan^2 x], then which of the following is true ?

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