Compared with the continuous-energy Monte Carlo neutron-transport calculation method, the majority of deterministic methods rely on the multi-group approximation. On one hand, it necessitates the establishment of a dedicated multi-group data library, which has to be obtained based on an approximate representation of the typical energy spectrum. On the other hand, resonance self-shielding calculation has to be done as a priori to obtain the effective multi-group cross-section in the resonance energy range considering the self-shielding effects in both space and energy domains, as well as interference effects. Consequently, a continuous-energy deterministic method had been proposed by expanding the neutron flux spectrum using different basis functions for different energy ranges. It has been proved to be capable of getting rid of the two problems caused by the multi-group approximation. However, it was limited in energy space only. Thus, in this paper, a continuous-energy Method of Characteristics (MOC) is proposed to extend the continuous-energy deterministic method in handle spatial and angular heteronomies. In this continuous-energy MOC, the continuous neutron-transport equation is first expanded by using energy basis functions to obtain the differential-integral equation in terms of the energetic expansion moments. Secondly, the cross-coupled moments equation is decomposed through eigen-decomposition of the total cross-section matrix, yielding a set of first-order nonhomogeneous ordinary differential equations. Thirdly, these equations can be projected onto the characteristic trajectory. Their general solutions can be derived according to the legacy MOC. The actual angular flux can then be reconstructed by using the eigen-vectors of the total cross-section matrix. In the full paper, the theoretical model would be derived in detail, while the possibility of this method will also be demonstrated through a series of verification results.