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We study the Steklov spectral problem for the Laplace operator in a boundeddomain $\Omega \subset \mathbb{R}^d$, $d \geq 2$, with a cusp such that thecontinuous spectrum of the problem is non-empty, and also in the family ofbounded domains $\Omega^\varepsilon \subset \Omega$, $\varepsilon > 0$,obtained from $\Omega$ by blunting the cusp at the distance of $\varepsilon$from the cusp tip. While the spectrum in the blunted domain$\Omega^\varepsilon$ consists for a fixed $\varepsilon$ of an unboundedpositive sequence $\{ \lambda_j^\varepsilon \}_{j=1}^\infty$ of eigenvalues, wesingle out different types of behavior of some eigenvalues as $\varepsilon \to+0$: in particular, stable, blinking, and gliding families of eigenvalues arefound. We also describe a mechanism which transforms the family of theeigenvalue sequences into the continuous spectrum of the problem in $\Omega$,when $\varepsilon \to +0$.

"Blinking eigenvalues" of the Steklov problem generate the continuous spectrum in a cuspidal domain