I'm a bit confused about something that appears in the fourth chapter of Hubbard's Teichmüller Theory text. In his statement of Weyl's lemma, he says that if $f$ is a distribution whose weak/distributional derivative $\frac{\partial f}{\partial \overline{z}}$ is zero, then $f$ is an analytic function. My interpretation of this statement would be that the weak partial derivative $\frac{\partial f}{\partial \overline{z}}$ is equal to zero up to a set of measure zero, since we're specifically talking about the distributional derivative. (I suppose that once we know $f$ is analytic this becomes upgraded to the standard partial derivative with respect to $\overline{z}$.) He then goes on to construct an example of a non-analytic function in the plane, $(x, y) \mapsto (x, y + \eta(x))$ where $\eta$ is the devil's staircase function, and mentions this function has partial derivative wrt $\overline{z}$ which is equal to zero almost everywhere.
The two statements by themselves make sense, but I'm a bit puzzled about why this example doesn't contradict Weyl's lemma if we are really talking about weak derivatives which are only defined up to measure zero.
Hubbard further goes on to mention that the issue of having "part of the distributional derivative hiding in a set of measure zero" (not an actual quote since I don't have the book in front me right now) is problematic, but I don't understand what this statement can mean if weak derivatives are only defined up to measure zero.