= Discrete integration by parts
{title2=$\int g\,\delta_h f=-\int(\delta_{-h}g)f$}
With $\delta_h f(x)=[f(x+he_\ell)-f(x)]/h$, a change of variables gives
$$
\int g\,\delta_h f=-\int(\delta_{-h}g)f.
$$
The <compact support> of $f$ and its translates must remain inside the integration domain, or one must integrate over the whole space with convergent integrals. In particular $\delta_{-h}g(x)=[g(x)-g(x-he_\ell)]/h$: the denominator's sign matters. This is the <difference quotient> analogue of <integration by parts>.
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