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

If the function $$f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in \mathbf{R}$$, is continuous at $$x=0$$, then $$f(0)$$ is equal to :

If the function

$$f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$$

is continuous at $$x=0$$, then the value of $$a^2$$ is equal to

Let $$f: \mathbf{R} \rightarrow \mathbf{R}$$ be a function given by

$$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0\end{cases}$$

where $$\alpha, \beta \in \mathbf{R}$$. If $$f$$ is continuous at $$x=0$$, then $$\alpha^2+\beta^2$$ is equal to :