Q223 / 270 · NDA 2016 (Phase 2) · Mathematics

Let $f: A \rightarrow R$, where $A=R \{0\}$ is such that $f(x)=\frac{x+|x|}{x}$. On which one of the following sets is $f(x)$ continuous?

  • Correct answer
    A
    A
  • B
    $B=\{x \in R: x \geq 0\}$
  • C
    $C=\{x \in R: x \leq 0\}$
  • D
    $\mathrm{D}=\mathrm{R}$

Solution

For $x \geq 0$ $f(x)=\frac{x+x}{x}=2$ For $x \lt 0$ $\begin{aligned} & f(x)=\frac{x-x}{x}=0 \\ & \lim _{x \rightarrow 0^{+}} f(x)=2 \\ & \lim _{x \rightarrow 0^{-}} f(x)=0 \\ & f(0)=2 \end{aligned}$ $\Rightarrow$ It is discontinuous at $x=0$. Option (a) is correct.

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

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