Question

In the following [x] denotes the greatest integer less than or equal to x

Answer 1

A) $f\left(x\right)=x\left|x\right|=\{\begin{array}{cc}-x^2& -1<x<0\\ x^2& 0\le x<1\end{array}$
Hence $f\left(x\right)$ is continuous and differentiable in $(-1,1)$
B) $f\left(x\right)=\sqrt{\left|x\right|}=\{\begin{array}{c}\sqrt{-x}-1<x<0\\ \sqrt{x}0\le x<1\end{array}$
Hence $f\left(x\right)$ is continuous on $(-1,1)$ but not diiferentialbe at $x=0$ (sharp point).
C) $f\left(x\right)=x+\left[x\right]=\{\begin{array}{c}x-1-1<x<0\\ x0\le x<1\end{array}$
Is strictly increasing for $(-1,1)$ but not continuous nor differentiable on $(-1,1)$
D) $f\left(x\right)=|x-1|+|x+1|=2$ for all $xϵ(-1,1)$ so it is continuous and differentiable in same interval.