The definition my lecturer gave for a complex function was that it is one whose domain and/or range is a subset of $\mathbb{C}$ that is not a subset of $\mathbb{R}$.
But according to this book, any real-valued function of a real variable is a complex function. This seems a little strange to me, because if that were the case then wouldn't it suggest all results in complex analysis also hold in real analysis (which of course they don't)?