There is a Van Douwen MAD family

We answer a long-standing question of Van Douwen by proving in $\mathrm{ZFC}$ that there is a MAD family of functions in ${\omega}^{\omega}$ that is also maximal with respect to infinite partial functions. In Section 3 we apply the idea of trace introduced in this proof to the still open question of whether analytic MAD families exist in ${\omega}^{\omega}$. Using the idea of trace, we show that any analytic MAD families that may exist in ${\omega}^{\omega}$ must satisfy strong combinatorial constraints. We also show that it is consistent to have MAD families in ${\omega}^{\omega}$ that satisfy these constraints.