Definition of spectrum of $\mathcal{L}$ is continuous

Question

What does it mean for the spectrum of eigenvalues of differential operator $\mathcal{L}$ to be anywhere continuous? The textbook that I'm using doesn't give the definition of a spectrum either. This is a textbook in Multivariable Calculus, not functional analysis (which I haven't learned). All the definitions I've seen have to do strongly with functional analysis.

Answer 1

I think they mean the following. Continuous spectrum of $\mathcal L$ is the subset of all those $\lambda\in\mathbb K$ for which $\mathcal L - \lambda I$ is injective, is not surjective and has dense image, where $I$ is the identity operator.