# Iterated integral does not exist but the function is integrable.

I'm reading Buck's Advanced calculus, and on Page 186, it says, if $$f$$ is bounded and continuous on a rectangle $$R$$ except at points of a set $$E$$ of zero area, it may happen that for individual values of $$x$$, $$\int^d_c f(x,y)dy$$ does not exist, and the set of $$x$$ for which this is true can be fail to be a set of zero length.
I'm wondering does that mean iterated integral of function $$f$$ does not exist but the function is integrable? And is there a concrete example such that a function is integrable but the iterated integral does no exist? Thanks!

Consider the function $$f(x,y)$$ given by $$\begin{cases} x^2y^2, &x\neq2 \\ \text{undefined}, &x = 2\\ \end{cases}$$ Since the problem is only along an infinitesimal slice of the domain, the function is integrable, and yet the iterated integral over $$y$$ when $$x=2$$ does not exist.