The function f (x) = x − [x], where [⋅] denotes

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.

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