$f$ is continuous at $x=\frac{\pi}{2}$ where,
$f(x)=\begin{cases}\frac{2 k \cos x}{\pi-2 x}, & x \neq \frac{\pi}{2} \\ 2024, & x=\frac{\pi}{2}\end{cases}$ then,the value of $k$ is . . . . . .

At $x=0$,the function $f(x) = \begin{cases} \frac{x}{|x|+2x^2}, & x \neq 0 \\ k, & x=0 \end{cases}$ is:

If $f(x) = \begin{cases} \frac{\sin 3x}{e^{2x}-1} & x \neq 0 \\ k-2 & x=0 \end{cases}$ is continuous at $x=0$,then $k=$

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

Let $f$ be a differentiable function on the open interval $(a, b)$. Which of the following statements must be true?
$I$. $f$ is continuous on the closed interval $[a, b]$
$II$. $f$ is bounded on the open interval $(a, b)$
$III$. If $a < a_1 < b_1 < b$,and $f(a_1) < 0 < f(b_1)$,then there is a number $c$ such that $a_1 < c < b_1$ and $f(c) = 0$

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