Translation of a distribution (${\tau }_{h}u$)

If ${\tau }_{h}f(x)=f(x-h)$ for functions, its extension to distributions is defined by

A distribution is invariant under every translation parallel to the $i$th coordinate axis exactly when its $i$th distributional derivative vanishes. Differentiate the translated pairing for one direction; integrate that derivative in the translation parameter for the converse.