Let $f:(1,\infty)\to(0,\infty)$ be a continuous function such that for every $n\in\mathbb{N}$, $f(n)$ is the smallest prime factor of $n$. Then which of the following is/are correct?
a) $lim_{x\to\infty} f(x)$ exists.
b) $lim_{x\to\infty} f(x)$ does not exists.
c) The set of solutions to the equation $f(x)=2024$ is finite.
d) The set of solutions to the equation $f(x)=2024$ is infinite.
$\textbf{My approach:}$ as the range set (for $n\in\mathbb{N}$) is $\{ 1,2,3,2,5,2,7,2,3,\cdots\}$ then $lim_{x\to\infty}$ does not exist.but for $f(x)=2024$ the set of solutions is finite as range can never get the value 2024 so it is empty set and empty set is finite. But it is given that set of solution to $f(x)=2024$ is infinite.
Please do check my approach and provide solution