(C) The function $f(x) = x - [x]$ is known as the fractional part function,denoted by $\{x\}$.By definition,for any integer $n$,we have $[x + n] = [x]

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$A$ periodic function whose period is $1$

(C) The function $f(x) = x - [x]$ is known as the fractional part function,denoted by $\{x\}$.By definition,for any integer $n$,we have $[x + n] = [x] + n$.Therefore,$f(x + 1) = (x + 1) - [x + 1] = x + 1 - ([x] + 1) = x - [x] = f(x)$.Since $f(x + 1) = f(x)$ for all $x \in \mathbb{R}$,the function is periodic with period $1$.

Class 12 Mathematics Relation and Function Functional Equations

Easy

The function $f(x) = x - [x]$,where $[x]$ denotes the greatest integer function,is:

A
$A$ periodic function
B
$A$ periodic function whose period is $\frac{1}{2}$
C
$A$ periodic function whose period is $1$
D
Not a periodic function

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DifficultJEE MAIN 2023

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Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy, \forall x, y \in \mathbb{R}$,then $\sum_{n=1}^{10} f(n)$ is equal to:

DifficultJEE MAIN 2017

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The function $f(x) = x - [x]$,where $[x]$ denotes the greatest integer function,is:

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$A$ function $f: R \rightarrow R$ satisfies $f\left(\frac{x+y}{3}\right) = \frac{f(x)+f(y)+f(0)}{3}$ for all $x, y \in R$. If the function $f$ is differentiable at $x=0$, then $f$ is:

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