In this paper we consider quasiequilibrium and quasioptimization problems. A relaxed level closedness notion is proposed and used together with pseudocontinuity to establish sufficient conditions for a family of such problems being (uniquely) well-posedness at the reference point. We prove them in general formulations, though such relaxations allow us to improve some existing results even in simple cases of R. For topological settings we use sensitivity analysis while for problems on metric spaces we argue on diameters and measures of noncompactness of approximate solution sets.