Question

The function $f\left(x\right)=max\{(1-x),(1+x),2\},xϵ(-\infty ,\infty )$ is-

• A. continuous at all points
• B. differentiable at all points
• C. differentiable at all points except at $x=1$ and $x=-1$
• D. continuous at all points except at $x=1$ and $x=-1$ where it is discontinuous

Answer 1

Answer: (A), (C)

The correct options are
A continuous at all points
C differentiable at all points except at $x=1$ and $x=-1$

$f\left(x\right)=max.\{(1-x),(1+x),2\}atx\in (-\infty ,\infty )$
Its clear from the graph that $f\left(x\right)$ is continuous at all points.
Also, graph takes sharp turns at $x=1,-1$. So $f\left(x\right)$ is not differentiable at $x=1,-1$.
So $f\left(x\right)$ is continuous at all points and differentiable at all points except $x=1$ and $x=-1$