Improper integrals with several singular ends

If $f$ has a possible singularity at $c\in(a,b)$, define $\int_a^b f:=\int_a^c f+\int_c^b f$ only when the two one-sided improper integrals converge separately. Cancellation between divergent sides is not allowed.

Likewise, for any finite split point $s$, define $\int_{-\infty}^{\infty}f:=\int_{-\infty}^{s}f+\int_s^\infty f$ only when both tails converge. More generally, an interval with several singular ends is split into finitely many one-ended pieces, each of which must converge separately. Independence of the permitted split point is a theorem, not part of this definition.