Domain: open set in the complex plane. (interior of a closed curve)

Region: Domain plus its boundary. (interior of a closed curve plus its boundary (the curve))

There are references to functions analytic in a domain (no problem), and analytic in a region, which would require f'(z) be defined at a boundary, but I can't find any definition for f'(z) at a boundary. Does such a definition exist? It would be analagous to the definition of a one-sided derivative of a function of a real variable.