If $f(x) = \begin{cases} x+a, & x \leq 0 \\ |x-4|, & x > 0 \end{cases}$ and $g(x) = \begin{cases} x+1, & x < 0 \\ (x-4)^2+b, & x \geq 0 \end{cases}$ are continuous on $\mathbb{R}$,then $(g \circ f)(2) + (f \circ g)(-2)$ is equal to.

If $f(x) = \frac{x^2-10x+25}{x^2-7x+10}$ and $f$ is continuous at $x=5$,then $f(5)$ is equal to

If the function $f: R \rightarrow R$ defined by $f(x) = \begin{cases} \frac{\sin(a + 1)x + \sin x}{x}, & x < 0 \\ b, & x = 0 \\ \frac{\sqrt{x + x^2} - \sqrt{x}}{x^{3/2}}, & x > 0 \end{cases}$ is continuous on $R$,then $a + b =$

Consider the function $f(x) = [x] + |1 - x|$ for $-1 \le x \le 3$,where $[x]$ is the greatest integer function.
Statement $1$: $f$ is not continuous at $x = 0, 1, 2$ and $3$.
Statement $2$: $f(x) = \begin{cases} -1 - x, & -1 \le x < 0 \\ 1 - x, & 0 \le x < 1 \\ 1 - x, & 1 \le x < 2 \\ 2 + x - 2, & 2 \le x < 3 \\ 3, & x = 3 \end{cases}$ (Note: The provided Statement $2$ in the prompt is incorrect).

Let $f, g: R \rightarrow R$ be functions defined by $f(x) = \begin{cases} x \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$ and $g(x) = x f(x)$. Consider the following statements: $(i)$ $f(x)$ is continuous at $x = 0$ but not differentiable at $x = 0$. $(ii)$ $g(x)$ is differentiable at $x = 0$, but $g'(x)$ is not continuous at $x = 0$. Then, which one of the following is true?

If the function $f$ defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $f(x)=\begin{cases} \frac{\sqrt{2} \cos x-1}{\cot x-1}, & x \neq \frac{\pi}{4} \\ k, & x=\frac{\pi}{4} \end{cases}$ is continuous,then $k$ is equal to