Deep learning-based dMRI super-resolution methods can effectively enhance image resolution by leveraging the learning capabilities of neural networks on large datasets. However, these methods tend to learn a fixed scale mapping between low-resolution (LR) and high-resolution (HR) images, overlooking the need for radiologists to scale the images at arbitrary resolutions. Moreover, the pixel-wise loss in the image domain tends to generate over-smoothed results, losing fine textures and edge information. To address these issues, we propose a novel continuous super-resolution of dMRI with anatomical structure-assisted implicit neural representation learning method, called CSR-dMRI. Specifically, the CSR-dMRI model consists of two components. The first is the latent feature extractor, which primarily extracts latent space feature maps from LR dMRI and anatomical images while learning structural prior information from the anatomical images. The second is the implicit function network, which utilizes voxel coordinates and latent feature vectors to generate voxel intensities at corresponding positions. Additionally, a frequency-domain-based loss is introduced to preserve the structural and texture information, further enhancing the image quality. Extensive experiments on the publicly available HCP dataset validate the effectiveness of our approach. Furthermore, our method demonstrates superior generalization capability and can be applied to arbitrary-scale super-resolution, including non-integer scale factors, expanding its applicability beyond conventional approaches.