There is a Van Douwen MAD family

We prove in ZFC that there is a MAD family of functions in omega^omega which is also maximal with respect to infinite partial functions. This solves a 20 year old question of Van Douwen. We also strengthen a result of J. Steprans stating that strongly MAD families of functions cannot be analytic. We show that analytic MAD families of functions, if they exist, must satisfy some strong constraints.