The Fourier transform can be defined on $L^2$ by
$$ \tilde f = \lim_{n\to\infty}\left(\omega\mapsto\int_{-n}^n f(x)\exp(i\omega x)dx\right),\tag1$$
where the limit is taken in $L^2$. But textbooks always take care to point out that this is the limit of the functions, rather than the 'function of the limits':
$$\omega\mapsto\lim_{n\to\infty}\int_{-n}^n f(x)\exp(i\omega x)dx.\tag2$$
What would go wrong if you were to use $(2)$ as the definition? Is there some $f$ for which $(2)$ doesn't converge everywhere, or converges to the wrong thing on a set of positive measure?