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This paper mainly concerns the frequency-preserving Kolmogorov-Arnold-Moser(KAM) theorem via irregular continuity with respect to the parameter. Insteadof digging out domains or requiring the uniform weak convexity for thefrequency mapping, we introduce the concept of relative singularity, allowingmany explicit parameterized Hamiltonian systems that admit arbitrarily weakregularity. The KAM iterative scheme employed here is quasi-linear that differsfrom the usual linear one. Additionally, we also construct a number ofcounterexamples, aiming to emphasize the indispensability of our new conditionstowards frequency-preserving. Moreover, we also show the parallel applicabilityof our results to partial frequency-preserving and infinite-dimensional cases.

A sharp frequency-preserving KAM theorem with continuous dependence on parameters and several counterexamples