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Number of points where a function with greatest integer terms is not continuous or not differentiable

This type of question, asked 5 times: 2021 Paper 24, 25 Jul 2022, 29 Jul 2022, 6 Apr 2023, 24 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 24

    89
    Let [t]\lbrack t\rbrack denote the greatest integer ≤t\leq t. The number of points where the function
    f(x)=[x]∣x2−1∣+sin⁡(π[x]+3)−[x+1],x∈(−2,2)f(x) = \lbrack x\rbrack\left| x^{2}- 1 \right|+ \sin\left( \frac{\pi}{\lbrack x\rbrack + 3} \right)- \lbrack x+ 1\rbrack,x\in ( - 2,2)
    is not continuous is

  2. Question 2

    25 Jul 2022

    86
    Let f(x)={∣4x2−8x+5∣, if 8x2−6x+1≥0[4x2−8x+5], if 8x2−6x+1<0f(x) =\left\{ \begin{matrix} \left| 4x^{2}- 8x + 5 \right|,\text{~if~}8x^{2}- 6x + 1 \geq 0 \\ \left\lbrack 4x^{2}- 8x + 5 \right\rbrack,\text{~}\text{if~}8x^{2}- 6x + 1 < 0 \end{matrix} \right., where [α]\lbrack\alpha\rbrack denotes the greatest integer less than or equal to α\alpha. Then the number of points in R where ff is not differentiable is

  3. Question 3

    29 Jul 2022

    176
    If [t]\lbrack t\rbrack denotes the greatest integer ≤t\leq t, then number of points, at which the function f(x)=4∣2x+3∣+f(x) = 4|2x + 3| + 9[x+12]−12[x+20]9\left\lbrack x +\frac{1}{2} \right\rbrack- 12\lbrack x + 20\rbrack is not differentiable in the open interval (−20,20)( - 20,20), is

  4. Question 4

    6 Apr 2023

    81
    Let a∈Za \in Z and [t] be the greatest integer ≤t\leq t. Then the number of points, where the function f(x)=[af(x) = \lbrack a +13sin⁡x],x∈(0,π)+ 13\sin x\rbrack,x \in (0,\pi) is not differentiable, is

  5. Question 5

    24 Jan 2025

    132
    Let [x]\lbrack x\rbrack denote the greatest integer function, and let mm and nn respectively be the numbers of the points, where the function f(x)=[x]+∣x−2∣f(x) = \lbrack x\rbrack + |x - 2|, −2<x<3- 2 < x < 3, is not continuous and not differentiable. Then m+nm + n is equal to:

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