Scale arbitrary super-resolution based on implicit image function gains increasing popularity since it can better represent the visual world in a continuous manner. However, existing scale arbitrary works are trained and evaluated on simulated datasets, where low-resolution images are generated from their ground truths by the simplest bicubic downsampling. These models exhibit limited generalization to real-world scenarios due to the greater complexity of real-world degradations. To address this issue, we build a RealArbiSR dataset, a new real-world super resolution benchmark with both integer and non-integer scaling factors for the training and evaluation of real-world scale arbitrary super-resolution. Moreover, we propose a Dual-level Deformable Implicit Representation (DDIR) to solve real-world scale arbitrary super-resolution. Specifically, we design the appearance embedding and deformation field to handle both image-level and pixel-level deformations caused by real-world degradations. The appearance embedding models the characteristics of low-resolution inputs to deal with photometric variations at different scales, and the pixel-based deformation field learns RGB differences which result from the deviations between the real-world and simulated degradations at arbitrary coordinates. Extensive experiments show our trained model achieves state-of-the-art performance on the RealArbiSR and RealSR benchmarks for real-world scale arbitrary super resolution. The dataset and code are available at https://github.com/nonozhizhiovo/RealArbiSR.