Question

Let $f\left(x\right)=\{\begin{array}{c}\frac{x}{1+\left|x\right|},\left|x\right|\ge 1\\ \frac{x}{1-\left|x\right|}.\left|x\right|<1\end{array}$.Then which of the following statements is correct?

• A. $f$ is continuous but not differentiable on $R-\left\{-1,1\right\}$
• B. $f$ is both continuous and differentiable on $R-\{-1,1\}$
• C. $f$ is not differentiable on $R-\{-1,0,1\}$
• D. $f$ is not continuous on $R-\{-1,0,1\}$

Answer 1

The correct option is A $f$ is continuous but not differentiable on $R-\left\{-1,1\right\}$

${lim}_{x\to 1^{+}}f\left(x\right)=\frac{1}{2}$

${lim}_{x\to 1^{-}}f\left(x\right)=\infty$

so, function is neither continuous nor differentiable at $1$

same happens in case of $-1$

so it is continuous in $R-\{-1,1\}$