Single-image super-resolution (SR) with fixed and discrete scale factors has achieved great progress due to the development of deep learning technology. However, the continuous-scale SR, which aims to use a single model to process arbitrary (integer or non-integer) scale factors, is still a challenging task. The existing SR models generally adopt static convolution to extract features, and thus unable to effectively perceive the change of scale factor, resulting in limited generalization performance on multi-scale SR tasks. Moreover, the existing continuous-scale upsampling modules do not make full use of multi-scale features and face problems such as checkerboard artifacts in the SR results and high computational complexity. To address the above problems, we propose a scale-aware dynamic network (SADN) for continuous-scale SR. First, we propose a scale-aware dynamic convolutional (SAD-Conv) layer for the feature learning of multiple SR tasks with various scales. The SAD-Conv layer can adaptively adjust the attention weights of multiple convolution kernels based on the scale factor, which enhances the expressive power of the model with a negligible extra computational cost. Second, we devise a continuous-scale upsampling module (CSUM) with the multi-bilinear local implicit function (MBLIF) for any-scale upsampling. The CSUM constructs multiple feature spaces with gradually increasing scales to approximate the continuous feature representation of an image, and then the MBLIF makes full use of multi-scale features to map arbitrary coordinates to RGB values in high-resolution space. We evaluate our SADN using various benchmarks. The experimental results show that the CSUM can replace the previous fixed-scale upsampling layers and obtain a continuous-scale SR network while maintaining performance. Our SADN uses much fewer parameters and outperforms the state-of-the-art SR methods.