Recently, the transform-based low-rank tensor factorization (t-LRTF) has emerged as a promising tool for multi-dimensional data recovery. However, the discrete transforms along the third (i.e., temporal/spectral) dimension are dominating in existing t-LRTF methods, which hinders their performance in addressing temporal/spectral degeneration scenarios, e.g., video frame interpolation and multispectral image (MSI) spectral super-resolution. To break this barrier, we propose a novel Functional Transform-based Low-Rank Tensor Factorization (FLRTF), where the learnable functional transform is expressed by the implicit neural representation with positional encodings. The continuity brought by this function allows FLRTF to capture the smoothness of data in the third dimension, which will benefit the recovery of temporal/spectral degeneration problems. To examine the effectiveness of FLRTF, we establish a general FLRTF-based multi-dimensional data recovery model. Experimental results, including video frame interpolation/extrapolation, MSI band interpolation, and MSI spectral super-resolution tasks, substantiate that FLRTF has superior performance as compared with representative data recovery methods.