Let $f(x) = \begin{cases} x^p \sin \left( \frac{1}{x} \right) + x|x^3|, & x \neq 0 \\ 0, & x = 0 \end{cases}$. Then the complete set of values of $p$ for which $f''(x)$ is continuous at $x = 0$ is:

If $f(x) = \begin{cases} \frac{|x - a|}{x - a}, & x \neq a \\ 1, & x = a \end{cases}$,then:

The value of $p$ for which the function $f(x) = \begin{cases} \frac{(4^x - 1)^3}{\sin(\frac{x}{p}) \log(1 + \frac{x^2}{3})}, & x \ne 0 \\ 12(\log 4)^3, & x = 0 \end{cases}$ is continuous at $x = 0$ is:

Let $[x]$ denote the greatest integer less than or equal to $x$. Then $f(x) = \frac{1 + \sin([\cos x])}{\cos([\sin x])}$ is

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

The function $f(x) = |x - 24|$ is