Cauchy principal values at a finite singularity and on the real line

Let $f$ be properly Riemann integrable on compact subintervals of $[a,b]\setminus\{c\}$, where $a<c<b$. Its Cauchy principal value at $c$ is $\operatorname{PV}\!\int_a^b f:=\lim_{\varepsilon\downarrow0}\left(\int_a^{c-\varepsilon}f+\int_{c+\varepsilon}^bf\right),$ provided this coupled limit is finite.

For a locally Riemann-integrable function on the real line, define $\operatorname{PV}\!\int_{-\infty}^{\infty}f:=\lim_{R\to\infty}\int_{-R}^{R}f,$ again only for a finite limit. A principal value couples the two truncations; it does not assert that the two one-sided improper integrals converge separately.