Determine the continuity of a function

The following question appeared in a graduate-level entrance exam. In this question more than 1 option can be correct.

I determined that option A is incorrect since $p(x)$ and $q(x)$ are unbounded on $\mathbb R$.
For option B I think of a possible step function that could lie between these two polynomials or perhaps define $f(x)$ as$$f(x) = \begin{cases} p(x) & x\in \mathbb Q \\ q(x) & x\notin \mathbb Q \end{cases}$$with $q(x) \gt p(x) \,\,\, \forall x \in \mathbb R$. I don't know if this is correct. This one also ends up disproving option D.

Option C also doesn't look possible because $p(x), q(x)>0$$\forall x$.I am missing out on something here. I would highly appreciate a detailed explanation.