We study the Steklov spectral problem for the Laplace operator in a bounded domain Ω⊂R^d, d≥2, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains Ω^ε⊂Ω, ε>0, obtained from Ω by blunting the cusp at the distance of ε from the cusp tip. While the spectrum in the blunted domain Ω^ε consists for a fixed ε of an unbounded positive sequence {λ_j ^ε}_j=1 ^∞ of eigenvalues, we single out different types of behavior of some eigenvalues as ε→+0: in particular, stable, blinking, and gliding families of eigenvalues are found. We also describe a mechanism which transforms the family of the eigenvalue sequences into the continuous spectrum of the problem in Ω, when ε→+0.