Question

For the function $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{1-x}{|x-1|},& x<1\end{array}\\ \begin{array}{cc}1,& x=1\end{array}\\ \begin{array}{cc}{x}^2,& x>1\end{array}\end{array}$ which of the following is true?

• A. It is continuous at all points
• B. It is continuous at all points except at $x=1$
• C. It is differentiable at all points
• D. None of these

Answer 1

Answer: (A)

It is continuous at all points

To examine this function $x=1,$

$f\left(1\right)=1$

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

$\underset{x\to 1^{-1}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}\frac{1-x}{|x-1|}=1$

Hence, it is continuous at all points.