MCQ
The point $(s),$ at which the function $f$ given by $f(x)=\left\{\begin{array}{l}\frac{x}{|x|}, x<0 \\ -1, x \geq 0\end{array}\right.$ is continuous, is/are
- ✓$x \in R$
- B$x=0$
- C$x \in R-\{0\}$
- D$x=-1$ and $1$
✓
Answer
Correct option: A.
$x \in R$
We have, $f(x)=\left\{\begin{array}{ll}\frac{x}{|x|}, & x<0 \\ -1, & x \geq 0\end{array}\right.$
$\begin{array}{l} \Rightarrow f(x)=\left\{\begin{array}{cc} \frac{x}{-x}=-1, & x<0 \\ -1, & x \geq 0 \end{array}\right. \\\end{array}$
$\Rightarrow f(x)=-1 \forall x \in R$
$\Rightarrow f(x)$ is continuous $\forall x \in R$ as it is a constant function.
$\begin{array}{l} \Rightarrow f(x)=\left\{\begin{array}{cc} \frac{x}{-x}=-1, & x<0 \\ -1, & x \geq 0 \end{array}\right. \\\end{array}$
$\Rightarrow f(x)=-1 \forall x \in R$
$\Rightarrow f(x)$ is continuous $\forall x \in R$ as it is a constant function.
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