Question

If $f\left(x\right)=\left[x\right]+[x+\frac{1}{2}]$, where [.] denotes the greatest integer function, then

• A. $f\left(x\right)$ is continuous at $x=\frac{1}{2}$
• B. ${lim}_{x\to \frac{1}{2}+0}f\left(x\right)=1$
• C. $f\left(x\right)$ is discontinuous at $x=\frac{1}{2}$
• D. ${lim}_{x\to \frac{1}{2}-}f\left(x\right)=1$

Answer 1

Answer: (B), (C)

${lim}_{x\to \frac{1}{2}+0}f\left(x\right)=1$
$f\left(x\right)$ is discontinuous at $x=\frac{1}{2}$

$f\left(x\right)=\left[x\right]+[x+\frac{1}{2}]$

at $x\to {\frac{1}{2}}^{+},x+\frac{1}{2}=\mathrm{1.00...}$

$[x+\frac{1}{2}]=1$

${lim}_{x\to {\frac{1}{2}}^{+}}f\left(x\right)=0+1=1$

at $x\to {\frac{1}{2}}^{-},x+\frac{1}{2}=\mathrm{0.99...}$

$[x+\frac{1}{2}]=0$

${lim}_{x\to {\frac{1}{2}}^{-}}f\left(x\right)=0+0=0$

LHL$\ne$RHL $\therefore$Discontinuous