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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!

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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.

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