$A$ function $f: R \rightarrow R$ defined by $f(x) = \begin{cases} 2x+3, & x \leq \frac{4}{3} \\ -3x^2+8x, & x > \frac{4}{3} \end{cases}$ is

Let $f: R \rightarrow R$ be a function defined by
$f(x) = \begin{cases} x^2 \sin \left(\frac{\pi}{x^2}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$
Then which of the following statements is $TRUE$?

If $f(x)$ is the signum function,then in terms of $f(x)$,the constant function $g(x)=1, \forall x \in R$ is

Let $A = \{x \in R : x \text{ is not a positive integer}\}$. Define a function $f : A \to R$ as $f(x) = \frac{2x}{x - 1}$. Then $f$ is

If $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}$ is a function such that $|f(\alpha) - \alpha| \leqslant 1$ for all $\alpha \in \{1, 2, 3, 4\}$,then the total number of such functions is:

If $f(x) = \sin([\pi^2]x) - \sin([-\pi^2]x)$,where $[x]$ denotes the greatest integer function $\leq x$,then which of the following is not true?