Question

If $f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right],x\in R$, where $\left[x\right]$ denotes the greatest integer function, then :

• A. $\underset{x\to 4^{+}}{lim}f\left(x\right)$ exists but $\underset{x\to 4^{-}}{lim}f\left(x\right)$ does not exist.
• B. $\underset{x\to 4^{-}}{lim}f\left(x\right)$ exists but $\underset{x\to 4^{+}}{lim}f\left(x\right)$ does not exist.
• C. Both $\underset{x\to 4^{-}}{lim}f\left(x\right)$ and $\underset{x\to 4^{+}}{lim}f\left(x\right)$ exist but are not equal.
• D. $f$ is continuous at $x=4$.

Answer 1

The correct option is D $f$ is continuous at $x=4$.

$f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right]$

LHL at $x=4$
$\underset{x\to 4^{-}}{lim}f\left(x\right)=3-0=3$

RHL at $x=4$
$\underset{x\to 4^{+}}{lim}f\left(x\right)=4-1=3$

$f\left(4\right)=3$
$\therefore f$ is continuous at $x=4$