One of the first things about definite integration included in summaries is the linearity: $\int_a^b (\alpha f + \beta g)(x) \, \textrm{d}x = \alpha \int_a^b f(x) \,\textrm{d}x + \beta \int_a^b g(x) \, \textrm{d}x. \,$
Isn't this a bit clumsy since it puts the obvious problem of convergence in some complicated condition on $f$ and $g$?
For example the definite integral:
$\int_0^1 f(x)\cot(\pi x)\textrm{d}x$
diverges most of the time except if $f := B_n$ where $B_n$ is the n-th Bernoulli polynomial ($n>1$ and odd!).
If $n=3$ then $B_3(x)=x^3-\frac{3}{2}x^2-\frac{1}{2}x$. Define $f:=(x^3-\frac{3}{2}x^2)\cot(\pi x)$ and $g:=\frac{1}{2}x \cot(\pi x)$ then
$\int_0^1 (f + g)(x) \, \textrm{d}x \neq \int_0^1 f(x) \,\textrm{d}x + \int_0^1 g(x) \, \textrm{d}x. \,$ since the left side converges and the right side diverges!
Why is the linearity of definite integration so proudly displayed if it is so fragile?