  Using differential renormalization, we calculate the complete two-point
function of the background gauge superfield in pure N=1 Supersymmetric
Yang-Mills theory to two loops. Ultraviolet and (off-shell) infrared
divergences are renormalized in position and momentum space respectively. This
allows us to reobtain the beta function from the dependence on the ultraviolet
renormalization scale in an infrared-safe way. The two-loop coefficient of the
beta function is generated by the one-loop ultraviolet renormalization of the
quantum gauge field via nonlocal terms which are infrared divergent on shell.
We also discuss the connection of the beta function to the flow of the
Wilsonian coupling.
