Suppose that $f$ is continuous on $[a, b]$ and that $f(x)$ is an integer for each $x$ in $[a, b]$. Then in $[a, b]$

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?

Check the points where the constant function $f(x)=k$ is continuous.

If the function $f(x) = \begin{cases} 1+\cos x, & x \leq 0 \\ a-x, & 0 < x \leq 2 \\ x^2-b^2, & x > 2 \end{cases}$ is continuous everywhere,then $a^2+b^2=$

If $f(x) = \begin{cases} 6 \beta - 3 \alpha x, & \text{if } -4 \leq x < -2 \\ 4x + 1, & \text{if } -2 \leq x \leq 2 \end{cases}$ is continuous on $[-4, 2]$,then $\alpha + \beta = $

If $f(x) = \begin{cases} \frac{x-|x|}{x}, & x < 0 \\ b\left(\frac{5x^2+a}{x^2-3x+2}\right), & 0 \leq x \leq 1 \\ -14, & x \geq 3 \end{cases}$ is a continuous function on $R$,then $(a, b) =$