Question

The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
(a) continuous everywhere
(b) continuous at integer points only
(c) continuous at non-integer points only
(d) differentiable everywhere

Answer 1

(c) continuous at non-integer points only

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ f\left(x\right)=x-\left[x\right] \\ \mathrm{Consider}n\mathrm{be}\mathrm{an}\mathrm{integer}. \\ f\left(x\right)=x-\left[x\right]=\left\{\begin{array}{l}x-\left(n-1\right)n-1\le x<n\\ \begin{array}{l}0x=n\\ x-nn\le x<n+1\end{array}\end{array}\right. \\ \mathrm{Now}, \\ \left(\mathrm{LHL}\mathrm{at}x=n\right)=\underset{x\to n^{-}}{\lim }f\left(x\right)=x-\left(n-1\right)=x-n+1 \\ \left(\mathrm{RHL}\mathrm{at}x=n\right)=\underset{x\to n^{+}}{\lim }f\left(x\right)=x-\left(n\right)=x-n \\ \mathrm{As},\mathrm{LHL}\ne \mathrm{RHL}\mathrm{at}x=n \\ \mathrm{i}.\mathrm{e}.,\mathrm{given}\mathrm{function}\mathrm{is}\mathrm{not}\mathrm{continuous}\mathrm{at}n. \\ \mathrm{Now},n\mathrm{is}\mathrm{any}\mathrm{integer}. \\ \mathrm{Therefore},\mathrm{given}\mathrm{function}\mathrm{is}\mathrm{not}\mathrm{continuous}\mathrm{at}\mathrm{integers}. \end{gathered}$$

Therefore, given points are continuous at non-integer points only.