Let $f\left(x\right)=\left\{\begin{array}{ll}x\left[\frac{1}{x}\right]+x\left[x\right]& \mathrm{if} x\ne 0\\ 0& \mathrm{if} x=0\end{array}\right.$, (where, $\left[x\right]$ denotes the greatest integer function). Then, the correct statements is/are

A function $f:R\to R$ is defined as

 $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{a{x}^2+bx+c+e^{nx}}{1+c·e^{nx}}$, where $f$ is continuous on $R$, then

Let $f\left(x\right)=\left\{\begin{array}{ll}3x^2-4\sqrt{x}+1& \mathrm{for} x<1\\ ax+b& \mathrm{for} x\ge 1\end{array}\right.$.

 If $f\left(x\right)$ is continuous and differentiable for all real values in its domain, then

If $f\left(x\right)=\left\{\begin{array}{ll}\left[x\right]+\sqrt{\left\{x\right\}},& x<1\\ \frac{1}{\left[x\right]+{\left\{x\right\}}^2},& x\ge 1\end{array}\right.$, then

[where, $[·]$ and $\{·\}$ represent the greatest integer and fractional part functions respectively]

If $f$ is an even function such that $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h}$ has some finite non-zero value, then

Let $f\left(x\right)$ be differentiable for real $x$ such that

$f^{\text{'}}\left(x\right)>0$ on $\left(-\infty ,-4\right)$,

$f^{\text{'}}\left(x\right)<0$ on $\left(-4, 6\right)$ and

$f^{\text{'}}\left(x\right)>0$ on $\left(6, \infty \right)$.

If $g\left(x\right)=f\left(10-2x\right),$ then the value of $g^{\text{'}}\left(2\right)$ is

Number of points where function $f\left(x\right)$ defined as

 $f:\left[0, 2\mathrm{\pi }\right]\to R, f\left(x\right)=\left\{\begin{array}{l}3-\left|\cos x-\frac{1}{\sqrt{2}}\right|, \left|\sin x\right|<\frac{1}{\sqrt{2}}\\ 2+\left|\cos x+\frac{1}{\sqrt{2}}\right|, \left|\sin x\right|\ge \frac{1}{\sqrt{2}}\end{array}\right.$ is

 non-differentiable is