I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.

Let $X$ be an uncountable set and let $F$ be a collection of real-valued functions on $X$ with the following property:

For every countable set $A\subset X$ there is a function $\alpha_{A}:A\to(0,+\infty) $ such that for every $f\in F$ the set $\left\{\frac{f(x)}{\alpha_{A}(x)}, x\in A\right\}$ is bounded.

Does there exist a function $\alpha:X\to(0,+\infty)$ such that for every $f\in F$ the set $\left\{\frac{f(x)}{\alpha(x)}, x\in X\right\}$ is bounded?

If the answer is "no", can we remedy the situation by assuming that $X$ is a topological space satisfying some mild conditions and $F\subset C(X)$?