Let $f:N\to N$ be a function such that $f\left(m+n\right)=f\left(m\right)+f\left(n\right)$ for every $m, n\in N.$ If $f\left(6\right)=18$ then $f\left(2\right)·f\left(3\right)$ is equal to :

Let a function $f:R\to R$ be defined as, $f\left(x\right)=\left\{\begin{array}{ll}\sin x-e^x& \mathrm{if} x\le 0\\ a+\left[-x\right]& \mathrm{if} 0<x<1\\ 2x-b& \mathrm{if} x\ge 1\end{array}\right.$

Where $\left[x\right]$ is the greatest integer less than or equal to $x.$ If $f$ is continuous on $R$, then $\left(a+b\right)$ is equal to:

Let $f:R\to R$ be defined as $f\left(x\right)=\left\{\begin{array}{cl}\frac{x^3}{(1-\cos 2x{)}^2}{\log }_e\left(\frac{1+2x{e}^{-2x}}{{\left(1-x{e}^{-x}\right)}^2}\right)& , x\ne 0\\ \alpha & , x=0\end{array}\right.$

If $f$ is continuous at $x=0,$ then $\alpha$ is equal to:

Let $f:R\to R$ be a function defined as $f\left(x\right)=\left\{\begin{array}{ccc}3\left(1-\frac{\left|x\right|}{2}\right)& \text{if}& \left|x\right|\le 2\\ 0& \text{if}& \left|x\right|>2\end{array}\right..$

Let $g:R\to R$ be given by $g\left(x\right)=f(x+2)-f(x-2).$ If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

Let $f:R\to R$ be defined as

$f\left(x\right)=\left\{\begin{array}{ll}\frac{\lambda \left|x^2-5x+6\right|}{\mu \left(5x-x^2-6\right)}& x<2\\ e^{\frac{\tan (x-2)}{x-\left[x\right]}}& x>2\\ \mu & x=2\end{array}\right.$

where $\left[x\right]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x=2$, then $\lambda +\mu$ is equal to :

Let $f:[0,\infty )\to [0,\infty )$ be defined as $f\left(x\right)={\int }_0^x\left[y\right]dy$ where $\left[x\right]$ is the greatest integer less than or equal to $x$. Which of the following is true?