Let $f:[-1,2] \rightarrow \mathbb{R}$ be given by $f(x)=2x^2+x+[x^2]-[x]$,where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points where $f$ is not continuous is:

If $f(x) = \begin{cases} [x] + [-x], & x \ne 2 \\ \lambda, & x = 2 \end{cases},$ then $f$ is continuous at $x = 2,$ provided $\lambda$ is (where $[.]$ is the Greatest Integer Function).

Let $[t]$ represent the greatest integer not more than $t$. Then the number of discontinuous points of $f(x) = [x^{1/x}]$ in $(0, \infty)$ is

Let $f$ and $g$ be real-valued functions. If $\lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{[f(x)+7]^{2 / 3}}=\frac{7}{4}$, $\lim _{x \rightarrow 0} f(x)=1$ and $\lim _{x \rightarrow 0} g(x)=\alpha$, then $h(x)= \begin{cases} \sin (\alpha x), & 0 \leq x \leq \frac{\pi}{10} \\ \cos (2 \alpha x), & \frac{\pi}{10} < x \leq \frac{\pi}{5} \end{cases}$ is:

If $f(x) = \begin{cases} \frac{|x-2|}{x-2}, & x \neq 2 \\ 1, & x = 2 \end{cases}$,then which of the following statements is true?

If the function $f(x) = \begin{cases} \frac{\tan 4x \times \cos 3x}{x} & , x \neq 0 \\ k & , x = 0 \end{cases}$ is continuous at $x = 0$,then the value of $k$ is equal to . . . . . .