#### Need solution for RD Sharma maths class 12 chapter Continuity exercise 8.2 question 12

Discontinuous at all integral points.

Hint:

Greatest integer function is discontinuous at integral points.

Given:

$g(x)=x-[x]$

Explanation:

$g(x)=x-[x]$

It is defined at all integral points

let n be an integer.

Then,

\begin{aligned} &g(n)=n-[n]=0\\ &\text { The left hand limit of } \mathrm{f} \text { at } \mathrm{x}=\mathrm{n} \text { is }\\ &\lim _{x \rightarrow n^{-}} g(x)=\lim _{x \rightarrow n^{n}}(x-[x])=\lim _{x \rightarrow n^{-}}(x)-\lim _{x \rightarrow n^{-}}[x]\\ &=n-(n-1)=1 \end{aligned}

\begin{aligned} &\text { The right hand limit of } f \text { at } x=n \text { is }\\ &\lim _{x \rightarrow n^{+}} g(x)=\lim _{x \rightarrow n^{+}}(x-[x])=\lim _{x \rightarrow n^{+}}(x)-\lim _{x \rightarrow n^{+}}[x]\\ &=n-n=0 \end{aligned}

\begin{aligned} &\text { L.H. } L \neq R . H . L\\ &\therefore g \text { is not continious at } x=n \end{aligned}

Hence g is discontinious at all integral points.