Let R be an E_∞-ring, and let I ⊂ π₀R be a finitely generated ideal such that R is complete along I. This thesis studies localizing invariants arising from pairs of the form (R, I). Precisely, the pair (R, I) gives rise to a category Nuc_R, the category of nuclear R-modules: this category contains the usual category of R-modules, as well as many I-complete R-modules with continuous maps between them. We then study localizing invariants applied to such categories. In this context, a localizing invariant T is said to be continuous if T ( Nuc_R) = lim ←n T(R||Iⁿ). Efimov proved that algebraic K-theory is continuous. The main result of this thesis builds from the continuity of K-theory to prove the same for topological cyclic and Hochschild homology.