I have a sequence of functions $\{f_n(x)\}$ and each $f_n:[0,1] \cap \mathbb{Q} \to \mathbb{R}$. $\{f_n(x)\}$ converges point-wise to some $f(x)$ for each rational $x$ in $[0,1]\cap \mathbb{Q}$. I can also show that $\{f_n(x)\}$ is a positive non-increasing function in $n$ for each $x \in [0,1] \cap \mathbb{Q}$.
Also, the limiting function $f$ in $ [0,1] \cap \mathbb{Q}$ and can be seen to be 'continuous'. I am not sure how to explain what I mean by continuous for a function on $[0,1] \cap \mathbb{Q}$. For example, the you can consider the limiting function to be of form $f(x) = x$ defined on $[0,1] \cap \mathbb{Q}$.
What are the conditions I need to ensure uniform convergence of $\{f_n\}$ to $f$? I know that if these functions were defined on $[0,1]$ and if I could still show that the sequence is positive non-increasing for each $x \in [0,1]$ and the limiting function is continuous in $[0,1]$ then that would have been sufficient. However, I am not sure what to do in this case where the domain is $[0,1] \cap \mathbb{Q}$.
PS: I really do not know much about real analysis so simple answers preferred!