Define $f: R \rightarrow R$ by $f(x) = [x] + \sqrt{x - [x]}$ for $x \in R$,where $[x]$ denotes the greatest integer function. Then the set of points at which $f$ is continuous is

Let $f(x) = \begin{cases} x^3+8; x < 0 \\ x^2-4; x \ge 0 \end{cases}$ and $g(x) = \begin{cases} (x-8)^{1/3}; x < 0 \\ (x+4)^{1/2}; x \ge 0 \end{cases}$. Then the number of points,where the function $g \circ f$ is discontinuous,is ————

Let $f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x}$,$x \neq 0$. In order for $f(x)$ to be continuous at $x = 0$,$f(0)$ must be defined as:

If $f(x)=\begin{cases} \frac{x-3}{|x-3|}+a & , x<3 \\ a+b & , x=3 \\ \frac{|x-3|}{x-3}+b & , x>3 \end{cases}$ is continuous at $x=3$,then the value of $a-b$ is

Let $f :[0,1] \rightarrow \mathbb{R}$ and $g :[0,1] \rightarrow \mathbb{R}$ be defined as follows:
$f(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases}$
$g(x) = \begin{cases} 0 & \text{if } x \text{ is rational} \\ 1 & \text{if } x \text{ is irrational} \end{cases}$
Then:

If $f(x) = \frac{x - e^x + \cos 2x}{x^2}$ for $x \neq 0$ is continuous at $x = 0$,then which of the following is true? (Note: $[x]$ and $\{x\}$ denote the greatest integer and fractional part functions,respectively.)