Single image super-resolution (SISR) with deep convolutional neural networks has recently attracted increasing attention due to its potentials to generate rich details. To obtain better fidelity and visual quality, most of existing methods are of heavy design with the depth of network. However, the lack of highfrequency information in the deep network and the limit of non-local context relations of pixels remain the core issue. In addition, there are many arbitrary-scale image super-resolution tasks, so it is a focus that provides an effective and efficient model for super resolution with arbitrary-scale. To this end, we propose a Feature Preserving and Enhancing Network (FPEN) based on implicit representation, which is aims to preserving high-frequency information in deep network and enhancing non-local contextual features of pixels to output more realistic images at arbitrary scales. In particular, our proposed High Frequency Preserving Block (HFPB) can divide the features into high-frequency components and low-frequency components, and allocate more operations to high-frequency components to ensure that high-frequency components can be preserved in the network. Since high-frequency information contains the details and texture of the image, it can restore the finer details of the image. Moreover, Pixel Continuity Attention (PiCA) module our proposed utilizes visual cues observed from pixels to adaptively recalibrate the pixel range that needs attention in the image to better achieve feature enhancement and generate smoother images. Extensive experiments conducted on benchmark SISR models and datasets show that Feature Preserving and Enhancing Network can be employed for various SISR tasks with arbitrary-scale to obtain the better visual quality than other state-of-the-art SR algorithms.INDEX TERMS Image super resolution with arbitrary-scale, high frequency preservingsingle methods, non-local attention methods, implicit neural representation.