A continuum limit removes an ultraviolet cutoff while keeping specified physical observables finite. In renormalization-group language, the bare trajectory must approach an ultraviolet fixed point and the correlation length measured in cutoff units must diverge.
An asymptotically safe quantum field theory approaches a non-Gaussian ultraviolet fixed point with finitely many relevant directions. Those relevant coordinates are fixed by finitely many measurements.
Quantum triviality means that removal of the cutoff forces the renormalized interaction to vanish, leaving a free continuum theory. Keeping a nonzero interaction then requires treating the model as an effective theory with finite cutoff.
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