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
- A$x \in R$
- B$x=0$
- C$x \in R-\{0\}$
- D$x=-1$ and 1
✓
Answer
We have, $f(x)=\left\{\begin{array}{ll}\frac{x}{|x|}, & x<0 \\ -1, & x \geq 0\end{array}\right.$
\[\begin{array}{l}
\Rightarrow f(x)=\left\{\begin{array}{cc}
\frac{x}{-x}=-1, & x<0 \\
-1, & x \geq 0
\end{array}\right. \\
\Rightarrow f(x)=-1 \forall x \in R
\end{array}\]
$\Rightarrow f(x)$ is continuous $\forall x \in R$ as it is a constant function.
\[\begin{array}{l}
\Rightarrow f(x)=\left\{\begin{array}{cc}
\frac{x}{-x}=-1, & x<0 \\
-1, & x \geq 0
\end{array}\right. \\
\Rightarrow f(x)=-1 \forall x \in R
\end{array}\]
$\Rightarrow f(x)$ is continuous $\forall x \in R$ as it is a constant function.
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