Moment Generating Function or Characteristic Function

Question

I'm wondering what should be taught in a first course in probability and statistics, moment generating function (mgf) or characteristic function (cf)? When I learnt the course, mgf was taught to me. However, since cf has some advantages (such as its existence for all probability distributions, etc.) over mgf, I'm wondering why don't we just teach only cf in a first course in probability and statistics. One can think of teaching both mgf and cf. However, that's not always possible in a course containing both probability and statistics. Any suggesion in this connection is greatly appreciated.