Q83 / 170 · Manipal (MET) 2015 · Mathematics

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

  • A
    is continuous at $x=\frac{1}{2}$
  • Correct answer
    B
    is discontinuous at $x=\frac{1}{2}$
  • C
    $\lim _{x \rightarrow(1 / 2)^{+}} f(x)=2$
  • D
    $\lim _{x \rightarrow(1 / 2)^{-}} f(x)=1$

Solution

We have, $f(x)=[x]+\left[x+\frac{1}{2}\right]= \begin{cases}0, & \text { if } 0 \lt x \lt \frac{1}{2} \\ 1, & \text { if } x=\frac{1}{2} \\ 1, & \text { if } \frac{1}{2} \lt x \lt 1\end{cases}$ Clearly, $f(x)$ is discontinuous at $x=\frac{1}{2}$. Also, $\lim _{x \rightarrow 1 / 2^{-}} f(x)=0$ and $\lim _{x \rightarrow 1 / 2^{+}} f(x)=1$

Hence, the correct answer is option: 'B'.

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