The function $f(x) = p[x + 1] + q[x - 1],$ where $[x]$ is the greatest integer function,is continuous at $x = 1$ if:

Let $f: R \to R$ be a function defined as:
$f(x) = \begin{cases} 5, & x \le 1 \\ a + bx, & 1 < x < 3 \\ b + 5x, & 3 \le x < 5 \\ 30, & x \ge 5 \end{cases}$
Then $f$ is:

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 $f(x) = \begin{cases} \frac{|x - a|}{x - a}, & x \neq a \\ 1, & x = a \end{cases}$,then:

Show that the function $f$ given by $f(x) = \begin{cases} x^3 + 3, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases}$ is not continuous at $x = 0$.

If $f(x) = \begin{cases} \frac{(e^{ax}-1) \log(1+x)}{\sin^2 x} & \text{if } x > 0 \\ 2 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x < 0 \end{cases}$ is continuous at $x = 0$,then $\sqrt{b^2 - a^2} = $