For a causal function, each term in U(Z) will be smaller if Z is taken to be inside the circle |Z| < 1 rather than on the rim |Z|=1. Thus, convergence at Z=0 and on the circle |Z|=1 implies convergence everywhere inside the unit circle. So boundedness combined with causality means convergence in the unit circle.
Convergence at but not on the circle would refer to a causal function with infinite energy, a case of no practical interest. What function converges on the circle, at , but not at ? What function converges at all three places, , , and ?