Let $f\left(x\right)=\left\{\begin{array}{ll}\left[x\right]& x\notin I\\ x-1& x\in I\end{array}\right.$ (where, $[.]$ denotes the greatest integer function and

 $g\left(x\right)=\left\{\begin{array}{ll}\sin x+\cos x,& x<0\\ 1,& x\ge 0\end{array}.\right.$ Then for $f\left(g\left(x\right)\right)$ at $x=0$

If $f\left(x\right)=\left\{\begin{array}{ll}\sin \left({\cos }^{-1}x\right)+\cos \left({\sin }^{-1}x\right),& x\le 0\\ \sin \left({\cos }^{-1}x\right)-\cos \left({\sin }^{-1}x\right),& x>0\end{array}\right.$ then, at $x=0$

Let $f\left(x\right)=\underset{n\to \infty }{\lim }\sum _{r=0}^{n-1}\frac{x}{(rx+1)\left\{\right(r+1)x+1\}}$. Then

Let the function is differentiable at $x=1$.

 $f\left(x\right)=\left\{\begin{array}{ll}\underset{n\to \infty }{\lim }\frac{ax(x-1){\left(\cot \frac{\pi x}{4}\right)}^n+\left(p{x}^2+2\right)}{{\left(\cot \frac{\pi x}{4}\right)}^n+1},& x\in (0, 1)\cup (1, 2)\\ 0& , x=1\end{array}\right.$

 Then, the value of $|a+p|$ is

Let $f$ be a function with continuous second derivative and $f\left(0\right)=0, f^{\text{'}}\left(0\right)=0.$ Define a function $g$ by 

$g\left(x\right)=\left\{\begin{array}{ll}\frac{f\left(x\right)}{x},& x\ne 0\\ 0,& x=0\end{array}\right.$.
Then, which of the following statements is correct?

Let $f\left(x\right)$ be a function defined on $(-a, a)$ with $a>0$. Assume that $f\left(x\right)$ is continuous at $x=0$ and $\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(kx\right)}{x}=\alpha$,
where $k\in (0,1)$, then

If $f\left(x\right)=x^{\frac{1}{3}}{\left(x-2\right)}^{\frac{2}{3}}$ for all $x,$ then the domain of $f^{\text{'}}$ is: