Is the function defined by $f(x) = |x|$ a continuous function?

If $f(x) = \frac{4^{x-\pi} + 4^{\pi-x} - 2}{(x-\pi)^2}$ for $x \neq \pi$ is continuous at $x = \pi$,then $f(\pi) = k$. Find the value of $k$.

If the function $f(x)$,defined below,is continuous in the interval $[0, \pi]$,then find the values of $a$ and $b$.
$f(x) = \begin{cases} x + a\sqrt{2}(\sin x), & 0 \le x < \frac{\pi}{4} \\ 2x(\cot x) + b, & \frac{\pi}{4} \le x \le \frac{\pi}{2} \\ a(\cos 2x) - b(\sin x), & \frac{\pi}{2} < x \le \pi \end{cases}$

Let the function $f$ be defined by the equation $f(x) = \begin{cases} 3x & \text{if } 0 \le x \le 1 \\ 5 - 3x & \text{if } 1 < x \le 2 \end{cases}$,then:

Let $f(x)=\lim _{\theta \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in R$. Consider the following two statements: $(I)$ $f(x)$ is discontinuous at $x=1$. $(II)$ $f(x)$ is continuous at $x=-1$. Then,

Let $f(x) = [x]\sin \left( \frac{\pi}{[x + 1]} \right)$,where $[.]$ denotes the greatest integer function. The domain of $f$ is and the points of discontinuity of $f$ in the domain are