I was wondering if I could get some references regarding the following problem. I have a functional equation: $$f(x;\alpha) = (T \circ f)(x;\alpha)$$ where $x \in [0,1]$. This equation involves a parameter $\alpha$, but I can apply the Contraction Mapping Theorem and know that there is a unique continuous function satisfying the equation regardless of the value of $\alpha$.
What I aim to do is as follows: I want to show the everywhere-continuity of $f(x;\alpha)$ with respect to $\alpha$. However, I do not have the solution in closed form. What kinds of keywords should I look for? Any comments would be more than helpful.