$f(x) = \begin{cases} \frac{x-4}{|x-4|} + a, & x < 4 \\ a+b, & x=4 \\ \frac{x-4}{|x-4|} + b, & x > 4 \end{cases}$
If $f(x)$ given above is continuous at $x=4$,then find the values of '$a$' and '$b$'.

Let $a, b \in R, b \neq 0$. Define a function $f(x) = \begin{cases} a \sin \frac{\pi}{2}(x-1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases}$. If $f$ is continuous at $x = 0$,then $10 - ab$ is equal to ...... .

The function $f(x) = [x]$,where $[x]$ is the greatest integer function,is continuous at:

If the function $f(x)$ is continuous in $0 \leq x \leq \pi$,then the value of $2a+3b$ is where $f(x) = \begin{cases} x+a \sqrt{2} \sin x & \text{if } 0 \leq x < \frac{\pi}{4} \\ 2x \cot x + b & \text{if } \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a \cos 2x - b \sin x & \text{if } \frac{\pi}{2} < x \leq \pi \end{cases}$

If the real valued function $f(x) = \begin{cases} \frac{(4^x - 1)^4 \cot(x \log 4)}{\sin(x \log 4) \log(1 + x^2 \log 4)}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$, then $e^k = $

The function $f(x) = \frac{x^2 - 1}{x^3 - 1}$ is not defined at $x = 1$. If the function is continuous at $x = 1$,then the value of $f(1)$ will be: