Question

Find all the points of discontinuity of the greatest integer function defined by $f\left(x\right)=\left[\frac{x}{3}\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Answer 1

We have $\left[x\right]=I$

Let $c$ be any integer

$f$ is continuous at $x=c$ if LHL$=$RHL$=f\left(c\right)$

L.H.S$={lim}_{x\to c^{-}}f\left(x\right)={lim}_{x\to c^{-}}\left[\frac{x}{3}\right]=\frac{c-1}{3}$

R.H.S$={lim}_{x\to c^{+}}f\left(x\right)={lim}_{x\to c^{+}}\left[\frac{x}{3}\right]=\frac{c}{3}$

Since ${lim}_{x\to c^{+}}f\left(x\right)\ne {lim}_{x\to c^{-}}f\left(x\right)$

$\therefore f$ is discontinuous for all integers.