Let f:D(z_0, r)\{z_0}->C be analytic and bounded on a punctured open disk D(z_0, r), then lim_(z->z_0) f(z) exists, and the function defined by f^~ :D(z_0, r)->C f^~(z) = {f(z) | for z!=z_0 lim_(z'->z_0) f(z') | for z = z_0 auto right match is analytic.