We present a detailed study of a continuum random phase approximation approach to quasielastic electron-nucleus and neutrino-nucleus scattering. The formalism is validated by confronting ($e,e$) cross-section predictions with electron scattering data for the nuclear targets $^{12}$C, $^{16}$O, and $^{40}$Ca, in the kinematic region where quasielastic scattering is expected to dominate. We examine the longitudinal and transverse contributions to $^{12}$C($e,e$) and compare them with the available data. Further, we study the $^{12}$C($ u_{mu},mu^{-}$) cross sections relevant for accelerator-based neutrino-oscillation experiments. We pay special attention to low-energy excitations which can account for non-negligible contributions in measurements, and require a beyond-Fermi-gas formalism.
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