Question

Let $f\left(x\right)=\{\begin{array}{cc}max\{e^x,e^{-x},2\},& x\le 0\\ min\{e^x,e^{-x},2\},& x>0\end{array}$.
Which of the following statements is (are) CORRECT?

• A. $f\left(x\right)$ is discontinuous at $x=0$.
• B. $f\left(x\right)$ is non-differentiable at exactly two points.
• C. $f\left(x\right)$ has non-removable type of discontinuity at $x=0$ with jump of discontinuity equal to $2$.
• D. $f\left(x\right)$ is continuous but non-differentiable at $x=\ln \frac{1}{2}$.

Answer 1

Answer: (A), (B), (D)

The correct options are
A $f\left(x\right)$ is discontinuous at $x=0$.
B $f\left(x\right)$ is non-differentiable at exactly two points.
D $f\left(x\right)$ is continuous but non-differentiable at $x=\ln \frac{1}{2}$.

$f\left(x\right)=\{\begin{array}{cc}{e}^{-x},& x<-\ln 2\\ 2,& -\ln 2\le x\le 0\\ e^{-x},& x>0\end{array}$