If the function $f(x) = \begin{cases} [\tan(\frac{\pi}{4} + x)]^{\frac{1}{x}}, & x \neq 0 \\ K, & x = 0 \end{cases}$ is continuous at $x = 0$,then $K = ?$

If $f(x) = |x|/x$ for $x \neq 0$ and $1$ for $x = 0$,then the function is

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 = $

Prove that the function defined by $f(x) = \tan x$ is a continuous function.

Discuss the continuity of the function $f,$ where $f$ is defined by $f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ 0, & \text{if } 0 \le x \le 1 \\ 4x, & \text{if } x > 1 \end{cases}$ at $x=3.$

Let $[x]$ be the greatest integer less than or equal to $x$. At which of the following point$(s)$ is the function $f(x) = x \cos(\pi(x + [x]))$ discontinuous?
$[A]$ $x = -1$
$[B]$ $x = 0$
$[C]$ $x = 2$
$[D]$ $x = 1$