I currently have an equation of the form $$F(a(x),b(x),c(x),\lambda)=0$$ and am trying to determine whether I can implicitly express $\lambda\in\mathbb{C}$ as a function of $a(x),b(x),c(x):\mathbb{\Omega}\subset \mathbb{R}\to \mathbb{C}$ which are assumed to be continuously differentiable in the real sense. I believe in order to show this I would need to prove that $\partial_\lambda F\neq 0$ and then use the implicit function theorem, but are there any other assumptions that need to be made on the function $F$ here to obtain this result? Additionally, I am confused as to whether I should be considering complex differentiability here as opposed to real differentiability. Some clarification would be very much appreciated!
Note: The expression of $F(a(x),b(x),c(x),\lambda)$ unfortunately involves conjugates of $\lambda$ and $a(x),b(x)$.