The value of $f(0)$,so that the function $f(x) = \frac{(27 - 2x)^{1/3} - 3}{9 - 3(243 + 5x)^{1/5}}, (x \ne 0)$ is continuous,is given by

Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \alpha+\frac{\sin [x]}{x}, & x>0 \\ 2, & x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & x < 0 \end{cases}$. If $f$ is continuous at $x=0$,find the value of $\alpha + \beta$.

The set of points of discontinuity of the function $f(x) = x - [x]$,where $x \in R$,is:

Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval $(-2, 1)$,where the function $f(x) = |[x]| + \sqrt{x - [x]}$ is discontinuous,is $........$.

If $f(x) = \begin{cases} \frac{1 - [x]}{1 + x}, & x \ne -1 \\ 1, & x = -1 \end{cases}$,then the value of $f(|2k|)$ will be (where $[.]$ denotes the greatest integer function). Which of the following statements is true?

Let $f(x) = \begin{cases} (3 - \sin(1/x))|x|, & x \ne 0 \\ 0, & x = 0 \end{cases}$. Then at $x = 0$,$f$ has a