Let T be a continuous scalar-type spectral operator defined on a quasicomplete locally convex space X, that is, \(T=\int fdP\) where P is an equicontinuous spectral measure in X and f is a P-integrable function. It is shown that \(\sigma\) (T) is precisely the closed P-essential range of the function f or, equivalently, that \(\sigma\) (T) is equal to the support of the (unique) equicontinuous spectral measure Q defined on the Borel sets of the extended complex plane \({\mathbb{C}}^*\) such that \(Q(\{\infty \}=0\) and \(T=\int \lambda dQ(\lambda).\) This result is then used to prove a spectral mapping theorem; namely that \(g(\sigma (T))=\sigma (g(T))\) for any Q-integrable function \(g:{\mathbb{C}}^*\to {\mathbb{C}}^*\) which is continuous on \(\sigma\) (T). This is an improvement on previous results of this type [e.g., \textit{H. Schaefer}, Acta Math., 107, 125-173 (1962; Zbl 0112.343), and \textit{N. Dunford}: Functions, series and operators, Volumes I, II (Budapest, 1980), 391-422, Colloq. Math. Soc. János Bolyai No.35, North Holland (1983)] since it covers the case when \(g(\sigma (T))\setminus \{\infty \}\) is an unbounded set in \({\mathbb{C}}\), a phenomenon which occurs often for continuous operators T defined in non-normable spaces X.