2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/180669In this paper we prove that the four-point function of massive $\vp_4^4$-theory is continuous as a function of its independent external momenta when posing the renormalization condition for the (physical) mass on-shell. The proof is based on integral representations derived inductively from the perturbative flow equations of the renormalization group. It closes a longstanding loophole in rigorous renormalization theory in so far as it shows the feasibility of a physical definition of the renormalized coupling.23 pages; to appear in Rev. Math. Physics few corrections, two explanatory paragraphs addedMathematical PhysicsHigh Energy Physics - Theorytext