The value that should be assigned to $f(0)$ so that the function $f(x)=(x+1)^{\cot x}$ is continuous at $x=0$,is

Let $[x]$ represent the greatest integer not more than $x$. The discontinuous points of the function $f(x) = \frac{5+[x]}{\sqrt{11+[x]-6 \sqrt{2+[x]}}}$ lie in the interval

If $f(x) = \frac{x^2 - 10x + 25}{x^2 - 7x + 10}$ for $x \neq 5$ and $f$ is continuous at $x = 5$,then $f(5) = $

If the function $f(x) = \begin{cases} \frac{\cos ax - \cos 9x}{x^2}, & x \neq 0 \\ 16, & x = 0 \end{cases}$ is continuous at $x = 0$,then $a =$

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:

Let $f: R \rightarrow R$ be defined as $f(x) = \begin{cases} 2 \sin \left(-\frac{\pi x}{2}\right), & \text{if } x < -1 \\ |ax^2 + x + b|, & \text{if } -1 \leq x \leq 1 \\ \sin(\pi x), & \text{if } x > 1 \end{cases}$. If $f(x)$ is continuous on $R$,then $a + b$ equals ..... .