If $f$ is a continuous real-valued function defined on a closed interval $[a, b]$,then the range of the function is . . . . . .

Consider the function $f(x) = \begin{cases} \frac{x}{[x]} & \text{if } 1 \leqslant x < 2 \\ 1 & \text{if } x = 2 \\ \sqrt{6-x} & \text{if } 2 < x \leqslant 3 \end{cases}$ where $[x]$ denotes the greatest integer function. At $x = 2$,the function:

If $f(x) = \begin{cases} \frac{\sin x}{x} + \cos x, & x \ne 0 \\ 2, & x = 0 \end{cases}$,then which of the following is true?

If $f(x) = \begin{cases} \sin x, & \text{if } x \leq 0 \\ x^2+a^2, & \text{if } 0 < x < 1 \\ bx+2, & \text{if } 1 \leq x \leq 2 \\ 0, & \text{if } x > 2 \end{cases}$ is continuous on $\mathbb{R}$, then $a+b+ab = $

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x^2}, & \text{for } x \neq 0 \\ \lambda, & \text{for } x = 0 \end{cases}$ and if $f$ is continuous at $x = 0$,then $\lambda$ is equal to

$f(x) = \begin{cases} \frac{\log x}{x-1}, & \text{if } x \neq 1 \\ k, & \text{if } x=1 \end{cases}$ is continuous at $x=1$,then the value of $k$ is