The function defined by $f(x) = \begin{cases} \frac{x-4}{|x-4|} + a, & x < 4 \\ a + b, & x = 4 \\ \frac{x-4}{|x-4|} + b, & x > 4 \end{cases}$ is continuous at $x = 4$,if the values of $a$ and $b$ are:

Prove that the function $f(x) = x^{n}$ is continuous at $x = n$,where $n$ is a positive integer.

If $f(x) = \begin{cases} x \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$,then at $x = 0$ the function $f(x)$ is

If $\begin{aligned} f(x) &= \frac{4 \sin \pi x}{5 x} \text{ for } x \neq 0 \\ &= 2k \text{ for } x = 0 \end{aligned}$ is continuous at $x = 0$,then the value of $k$ is

If $f(x) = \begin{cases} mx+1, & x \leq \frac{\pi}{2} \\ \sin x+n, & x > \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$,where $m, n \in \mathbb{Z}$,then:

If the function $f(x) = \begin{cases} (1+|\cos x|)^{\frac{\lambda}{|\cos x|}} & , 0 < x < \frac{\pi}{2} \\ \mu & , x = \frac{\pi}{2} \\ e^{\frac{\cot 6x}{\cot 4x}} & , \frac{\pi}{2} < x < \pi \end{cases}$ is continuous at $x = \frac{\pi}{2}$,then $9\lambda + 6 \log_{e} \mu + \mu^6 - e^{6\lambda}$ is equal to