Question

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

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

Answer 1

Answer: (B)

is discontinuous at $x=\frac{1}{2}$

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

So, function is discontinuous at $x=\frac{1}{2}$.