I am reading a book (Reed and Simon vol I) on the continuous functional calculus and it speaks of a map $$ \phi_A:C(\sigma(A)) \to \mathcal{L}(H) $$ where$A$ is a self-adjoint operator on a Hilbert space $A$. In fact $\phi_A(P) = P(A)$ where $P(A) = \sum_{n=0}^\infty a_n A^n$. It then says that $\phi_A$ has a unique linera extension to the closure of the polynomials in $C(\sigma(A))$ and this closure is all of $C(\sigma(A))$.
But I don't understand how we are speaking of continuous functions on the spectrum of the operator $A$. What if its spectrum $\sigma(A)$ features discrete real numbers or gaps - how can we we have continuous functions defined on a set that may consist of discrete points or be the union of disjoint intervals with gaps?
If $\sigma(A)$ is discrete, then every function is continuous. You can always take the "subspace topology": make the topology of $\sigma(A)$ be $\{\sigma(A)\cap V:\ V\ \text{ is open}\}$. This works fine and gives a natural topology. And one can discuss continuity in any topology, it doesn't matter if it has "gaps". The only thing that changes is that it is now not very intuitive what it means to be continuous.