Examine whether the function $f$ given by $f(x) = x^{2}$ is continuous at $x = 0$.

The value of $k$ $(k > 0)$,for which the function $f(x) = \frac{(e^x - 1)^4}{\sin(\frac{x^2}{k^2}) \log(1 + \frac{x^2}{2})}$,where $x \neq 0$ and $f(0) = 8$,is continuous at $x = 0$,is

Discuss the continuity of the function $f,$ where $f$ is defined by $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}$ at $x=3.$

If in the interval $[0,3]$,$f(x) = \begin{cases} x\{x\}^2, & x \notin I \\ x, & x \in I \end{cases}$,then which of the following statements is correct? (where $\{.\}$ denotes the fractional part function)

If the function $f(x) = \frac{1-\sin 2x + \cos 2x}{1+\sin 2x + \cos 2x}$ for $x \neq \frac{\pi}{2}$ and $f(x) = k$ for $x = \frac{\pi}{2}$ is continuous at $x = \frac{\pi}{2}$,then $k = $

The number of discontinuities in $\mathbb{R}$ for the function $f(x)=\frac{x-1}{x^3+6x^2+11x+6}$ is