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 $f(x) = \frac{\log_{\sin |x|} \cos^3 x}{\log_{\sin |3x|} \cos^3 (x/2)}$ for $|x| < \frac{\pi}{3}, x \neq 0$ and $f(0) = 4$,then the number of points of discontinuity of $f$ in $\left( -\frac{\pi}{3}, \frac{\pi}{3} \right)$ is:

The function $f(x) = \frac{x+1}{9x+x^3}$ is

If $f(x) = \begin{cases} \frac{a + 3\cos x}{x^2}, & x < 0 \\ b\tan \left( \frac{\pi}{[x + 3]} \right), & x \geqslant 0 \end{cases}$ is continuous at $x = 0$,then:

If $f(x) = \begin{cases} \sin^{-1}|x|, & x \ne 0 \\ 0, & x = 0 \end{cases}$ then

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