Existence of mixed partials in Clairaut's theorem.

The fact under consideration is true. It amounts to Fubini's Theorem, in this shape: for some continuous $h(x,y)$ define $$ G(x,y) = \int_a^x \int_c^y \; h(u,v) \; dv \; du $$ Fubini says that we can interchange the order of integration. That and the fundamental theorem of calculus say that both $$ \frac{\partial}{\partial y} \left( \frac{\partial G}{\partial x} \right) $$ and $$ \frac{\partial}{\partial x} \left( \frac{\partial G}{\partial y} \right) $$ exist, are continuous and equal to $h.$

In particular, if $ \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) $ exists and is continuous, iterated integration tells us that $$ G(x,y) = f(x,y) - f(x,c) - f(a,y) + f(a,c), $$ with $$ \frac{\partial}{\partial y} \left( \frac{\partial G}{\partial x} \right) = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) $$ However, the other order exists for $G,$ so we get existence for $f$ and $$ \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial}{\partial x} \left( \frac{\partial G}{\partial y} \right) $$

This argument is from AKSOY_MARTELLI. I was able to view the pdf on my computer screen but not print it out. We are talking about the first part of Theorem 3, (i) implies (ii), bottom of page 128 to the first paragraph on page 129.