Recent Blind Image Super-Resolution (BSR) methods have shown proficiency in general images. However, we find that the efficacy of recent methods obviously diminishes when employed on image data with blur, while images with intentional blur constitute a substantial proportion of general data. To further investigate and address this issue, we developed a new super-resolution dataset specifically tailored for blur images, named the Real-world Blur-kept Super-Resolution (ReBlurSR) dataset, which consists of nearly 3000 defocus and motion blur image samples with diverse blur sizes and varying blur intensities. Furthermore, we propose a new BSR framework for blur images called Perceptual-Blur-adaptive Super-Resolution (PBaSR), which comprises two main modules: the Cross Disentanglement Module (CDM) and the Cross Fusion Module (CFM). The CDM utilizes a dual-branch parallelism to isolate conflicting blur and general data during optimization. The CFM fuses the well-optimized prior from these distinct domains cost-effectively and efficiently based on model interpolation. By integrating these two modules, PBaSR achieves commendable performance on both general and blur data without any additional inference and deployment cost and is generalizable across multiple model architectures. Rich experiments show that PBaSR achieves state-of-the-art (SOTA) performance across various quantitative metrics without incurring extra inference costs. Within the commonly adopted Learned Perceptual Image Patch Similarity (LPIPS) setting, PBaSR achieves an improvement range of approximately 0.02-0.09 with diverse anchor methods and blur types, across both the ReBlurSR and multiple widely used general BSR benchmarks.