This is a precalculus homework. (I'm afraid I could be missing something.)
Question. Let $p(x)$, $q(x)$ be functions defined in all points on a closed interval. Is the product $p(x)q(x)$ defined in all points on the closed interval? Is the quotient $p(x)/q(x)$ defined in all points in the closed interval?
Answer question 1. My answer is "yes" to the first question. How could I prove it, though? Here's my attempt. If both functions are defined everywhere on the interval, then in every point $x$ we have two real numbers $p(x)$, $q(x)$. If we multiply two real numbers, we get another. I think I must still argue that no division by zero will show up anywhere. Here's my attempt. The only possibility for us to have a denominator in the product $p(x)q(x)$ is if this denominator is already present in $p(x)$ or $q(x)$. But if that's so, we surely get both denominators different than zero --- otherwise one or both of the functions (individually) wouldn't be defined everywhere in the closed interval. (Is it only division by zero that's a danger?)
Answer question 2. My answer is "no" to the second question. The fact that both are defined everywhere on the interval says that all division by zero in each function individually is already excluded. However, say there is an $c$ in the interval such that $q(c) = 0$. Then $p(c)/q(c)$ is not defined. Therefore, the answer must be "no".