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Image super-resolution (SR) has attracted increasing attention due to its wide applications. However, current SR methods generally suffer from over-smoothing and artifacts, and most work only with fixed magnifications. This paper introduces an Implicit Diffusion Model (IDM) for high-fidelity continuous image super-resolution. IDM integrates an implicit neural representation and a denoising diffusion model in a unified end-to-end framework, where the implicit neural representation is adopted in the decoding process to learn continuous-resolution representation. Furthermore, we design a scale-controllable conditioning mechanism that consists of a low-resolution (LR) conditioning network and a scaling factor. The scaling factor regulates the resolution and accordingly modulates the proportion of the LR information and generated features in the final output, which enables the model to accommodate the continuous-resolution requirement. Extensive experiments validate the effectiveness of our IDM and demonstrate its superior performance over prior arts.