Let $f(x) = [x^2] \sin(\pi x)$, for $x > 0$. Then:

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:

If $f(x) = [x] - [\frac{x}{4}]$,$x \in R$,where $[x]$ denotes the greatest integer function,then

Assertion $(A)$: $f(x)=|x-a|+|x-b|$ is continuous on $R$. Reason $(R)$: $\frac{|x-\alpha|}{x-\alpha}$ is continuous at $x \in R-\{\alpha\}$. The correct option among the following is:

Suppose that $f$ is continuous on $[a, b]$ and that $f(x)$ is an integer for each $x$ in $[a, b]$. Then in $[a, b]$

Let $f:[0, \infty) \rightarrow [0, \infty)$ be defined as $f(x) = \int_{0}^{x} [y] \, dy$,where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?