Continuity of the four-point function of massive φ₄ ⁴-theory above threshold

In this paper we prove that the four-point function of massive φ₄⁴-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.