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
Solution:
(c) continuous at non-integer points only
We have,
$f(x)=x-[x]$
Consider $n$ be an integer.
$f(x)=x-[x]= \begin{cases}x-(n-1) & n-1 \leq x
Now,
$(\mathrm{LHL}$ at $x=n)=\lim _{x \rightarrow n^{-}} f(x)=x-(n-1)=x-n+1$
$(\mathrm{RHL}$ at $x=n)=\lim _{x \rightarrow n^{+}} f(x)=x-(n)=x-n$
As, $\mathrm{LHL} \neq \mathrm{RHL}$ at $x=n$
i.e., given function is not continuous at $n$.
Now, $n$ is any integer.
Therefore, given function is not continuous at integers.
Therefore, given points are continuous at non-integer points only.