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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2016 · Mathematics · Continuity and Differentiability

Let f: A arrow R , where A=R \ 0\ is such that f(x)=(x+|x|)/(x) ....

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

Options

  1. AA
  2. BB={x∈R:x≥0}B=\{x \in \mathbb{R}: x \geq 0\}
  3. CC={x∈R:x≤0}C=\{x \in \mathbb{R}: x \leq 0\}
  4. DD=R\mathrm{D}=\mathrm{R}

Correct answer

A. A

Step-by-step solution

For x≥0f(x)=x+xx=2x \geq 0f(x)=\frac{x+x}{x}=2 For x<0x < 0f(x)=x−xx=0lim⁡x→0+f(x)=2lim⁡x→0−f(x)=0f(0)=2\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=0x=0. Option (a) is correct.
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