Question

Show that the function defined by $g\left(x\right)=x-\left[x\right]$ is discontinuous at all integral point. Here $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Answer 1

Check the continuity of $g\left(x\right)$ at $x=a$ , where $a$ is an integer

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

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

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

$=\underset{h\to 0}{lim}h=0$

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

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

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

$=\underset{h\to 0}{lim}h+1=1$

$RHL\ne LHL$

So, the function is discontinuous at all integral point.