Our understanding of the human connectome is fundamentally limited by the resolution of diffusion MR images. Reconstructing a connectome's constituent neural pathways with tractography requires following a continuous field of fiber directions. Typically, this field is found with simple trilinear interpolation in low-resolution, noisy diffusion MRIs. However, trilinear interpolation struggles following fine-scale changes in low-quality data. Recent deep learning methods in super-resolving diffusion MRIs have focused on upsampling to a fixed spatial grid, but this does not satisfy tractography's need for a continuous field. In this work, we propose FENRI, a novel method that learns spatially-continuous fiber orientation density functions from low-resolution diffusion-weighted images. To quantify FENRI's capabilities in tractography, we also introduce an expanded simulated dataset built for evaluating deep-learning tractography models. We demonstrate that FENRI accurately predicts high-resolution fiber orientations from realistic low-quality data, and that FENRI-based tractography offers improved streamline reconstruction over the current use of trilinear interpolation.