Let $f : [0,1] \to [0,1]$ be a continuous function,then the equation $f(x) = x$

Given $f(x) = \begin{cases} \frac{\ln(1+\text{sgn}[x]+{x}^2)}{1-\cos{x}} & \text{if } x \neq 0 \\ k & \text{if } x = 0 \end{cases}$ (where $[\cdot]$,${\cdot}$ and $\text{sgn } x$ denote the greatest integer function,fractional part function,and signum function respectively),which of the following is true?

If the function $f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}$ is continuous at $x = 0$,then $f(0)$ is equal to . . . . . . .

If $f(x) = \begin{cases} \frac{5}{2} - x, & x < 2 \\ 1, & x = 2 \\ x - \frac{3}{2}, & x > 2 \end{cases}$,then:

Which one of the following functions is discontinuous at $x=1$?

Show that the function defined by $g(x)=x-[x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less than or equal to $x$.