This is probably trivial but for quite a few theorems dom(f) needs to be convex. However, often dom(f) is taken to be open. I know what these signify individually but was wondering if an open dom(f) is also considered convex or whether I would need to prove this as an additonal step to then use another theorem that requires dom(f) to be convex.
Not all open domains are convex. Take for example the subset of R^2 excluding the (closed) unit disk with radius <= 1. This domain is open, but not convex.
In principle, if a theorem requires a convex domain, you would have to show it, unless it's obvious (e.g. for all real numbers or a higher-dimensional Euclidian space)