We study the semilinear problem with the boundary reaction −∆u + u = 0 in Ω, ∂u ∂ν = λf(u) on ∂Ω, where Ω ⊂ R N , N > 2, is a smooth bounded domain, f : [0, ∞) → (0, ∞) is a smooth, strictly positive, convex, increasing function which is superlinear at ∞, and λ > 0 is a parameter. It is known that there exists an extremal parameter λ ∗ > 0 such that a classical minimal solution exists for λ < λ∗ , and there is no solution for λ > λ∗ . Moreover, there is a unique weak solution u ∗ corresponding to the parameter λ = λ ∗ . In this paper, we continue to study the spectral properties of u ∗ and show a phenomenon of continuum spectrum for the corresponding linearized eigenvalue problem.