Let $a\in C[0,1]$ and $A:C[0,1]\rightarrow C[0,1]$ be a bounded linear operator defined by $$A(x)(t)=a(t)x(t)\qquad \forall t\in [0,1].$$ Find the point, recidual and countinuous spectrum, that is, $\sigma_{p}(A)$, $\sigma_{c}(A)$, $\sigma_{r}(A)$. The total spectrum is noted by $\sigma(A)$.
My attempt: I showed that
$$\sigma_{p}(A):=\left\{\lambda \left|\: \exists U\subseteq [0,1] \mbox{ open such that } f|_{U}=\lambda \right.\right\}$$
Note that $A-\lambda I$ is not surjective iff $\lambda \in \mathrm{Imag}(a)$, therefore $ \mathrm{Imag}(a)= \sigma_{c}(A)\cup \sigma_{r}(A)$.
My problem: I have not been able to characterize $\sigma_{c}(A)$ and $\sigma_{r}(A)$.
Remark: It is important to clarify that I do not trust myself, so I would also like to know if the conclusions of my attempt are correct.