If the function given by $f(x) = \left(\frac{4x+1}{1-4x}\right)^{\frac{1}{x}}$ for $x \neq 0$ is continuous at $x = 0$,then the value of $f(0)$ is

If a real valued function $f(x) = \begin{cases} e^{\frac{\sin a(x-[x])}{x-[x]}}, & \text{if } x < 1 \\ b+1, & \text{if } x = 1 \\ \frac{|x^2+x-2|}{x-1}, & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$,then $b \sin a =$ ([x] denotes the greatest integer function)

If $f(x) = \lim_{n \to \infty} \frac{[x^2] + [(2x)^2] + [(3x)^2] + \cdots + [(nx)^2]}{n^3}$,then $f(x)$ is (Where $[\cdot]$ is the Greatest Integer Function).

If for $p \neq q \neq 0$,the function $f(x) = \frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$,then:

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 $x$.

Consider $f(x) = \begin{cases} \frac{x^2}{|x|}, & x \ne 0 \\ 0, & x = 0 \end{cases}$