(D) Given $f(x) = [x]^2 - [x^2]$.For $-1 < x < 0$,$[x] = -1$ and $[x^2] = 0$,so $f(x) = (-1)^2 - 0 = 1$.For $x = 0$,$f(0) = 0^2 - 0 = 0$.For $0 < x <

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(D) Given $f(x) = [x]^2 - [x^2]$.For $-1 For $x = 0$,$f(0) = 0^2 - 0 = 0$.For $0 For $x = 1$,$f(1) = 1^2 - 1 = 0$.For $1 For $x = \sqrt{2}$,$f(\sqrt{2}) = 1^2 - 2 = -1$.For $\sqrt{2} For $x = \sqrt{3}$,$f(\sqrt{3}) = 1^2 - 3 = -2$.For $\sqrt{3} For $x = 2$,$f(2) = 2^2 - 4 = 0$.For $2 For $x = \sqrt{5}$,$f(\sqrt{5}) = 2^2 - 5 = -1$.By analyzing the behavior at integers $n$,we find that the function is discontinuous at all integers except $x = 1$.

Class 12 Mathematics Continuity and Differentiation Continuity

Medium

The function $f(x) = [x]^2 - [x^2]$,(where $[y]$ is the greatest integer less than or equal to $y$),is discontinuous at

IIT 1999 All IIT

A
All integers
B
All integers except $0$ and $1$
C
All integers except $0$
D
All integers except $1$

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IIT 1999 Paper All IIT years →

Let $f:[-1,2] \rightarrow \mathbb{R}$ be given by $f(x)=2x^2+x+[x^2]-[x]$,where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points where $f$ is not continuous is:

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Find the values of $k$ so that the function $f$ is continuous at the indicated point. $f(x) = \begin{cases} kx^2, & \text{if } x \le 2 \\ 3, & \text{if } x > 2 \end{cases}$ at $x=2$.

Medium

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If $[x]$ denotes the greatest integer not exceeding the number $x$,then $f(x)$ defined by $f(x) = \begin{cases} [x], & \text{if } x < 2 \\ [x]-1, & \text{if } x \geq 2 \end{cases}$ is continuous in the interval.

MediumAP EAMCET 2018

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If $f$ is a continuous real-valued function defined on a closed interval $[a, b]$,then the range of the function is . . . . . .

EasyAP EAMCET 2020

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Prove that the identity function on real numbers given by $f(x) = x$ is continuous at every real number.

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The function $f(x) = [x]^2 - [x^2]$,(where $[y]$ is the greatest integer less than or equal to $y$),is discontinuous at

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Let $f:[-1,2] \rightarrow \mathbb{R}$ be given by $f(x)=2x^2+x+[x^2]-[x]$,where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points where $f$ is not continuous is:

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