Question

Find all the points of discontinuity of the greatest integer function defined by $f\left(x\right)=\left[x\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Answer 1

When $x$ is an integer

Given: function is $f\left(x\right)=\left[x\right]$

Let ${}^{\prime }{a}^{\prime }$ be an integer.

If $f\left(x\right)$ is continuous at $x=a$ then
$\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$

Finding L.H.L.

$\underset{x\to a^{-}}{lim}\left[x\right]$

$=\underset{h\to 0}{lim}[a-h]=(a-1)$

Finding R.H.L.

$\underset{x\to a^{+}}{lim}\left[x\right]$

$=\underset{h\to 0}{lim}[a+h]=a$

To find $f\left(x\right)$ at $x=a$
$f\left(x\right)=\left[x\right]\text{at}x=a$
$f\left(a\right)=\left[a\right]=a$

Hence, $f\left(x\right)$ is discontinuous at all integers.

when $x$ is not an integer

Let ${}^{\prime }{c}^{\prime }$ be the non-integer point and $c_1<c<c_2$
where $c_1,c_2\in Z$

If $f\left(x\right)$ is continuous at $x=c$ then

$\underset{x\to c^{-}}{lim}f\left(x\right)=\underset{x\to c^{+}}{lim}f\left(x\right)=f\left(c\right)$

Finding L.H.L.

$\underset{x\to c^{-}}{lim}\left[x\right]$

$=\underset{h\to 0}{lim}[c-h]=c_1$

Finding R.H.L.

$\underset{x\to c^{+}}{lim}\left[x\right]$

$=\underset{h\to 0}{lim}[c+h]=c_1$

To find $f\left(x\right)$ at $x=c$
$f\left(x\right)=\left[x\right]\text{at}x=c$
$f\left(c\right)=\left[c\right]=c_1$

Hence, ${lim}_{x\to c^{-}}f\left(x\right)={lim}_{x\to c^{+}}f\left(x\right)=f\left(c\right)$
Therefore, function $f\left(x\right)=\left[x\right]$ is discontinuous at all ‘integers’ but continuous at ‘non-integers’ points.