Object recognition using single-point supervision has attracted increasing attention recently. However, the performance gap compared with fully-supervised algorithms remains large. Previous works generated class-agnostic \textbf{\textit{proposals in an image}} offline and then treated mixed candidates as a single bag, putting a huge burden on multiple instance learning (MIL). In this paper, we introduce Point-to-Box Network (P2BNet), which constructs balanced \textbf{\textit{instance-level proposal bags}} by generating proposals in an anchor-like way and refining the proposals in a coarse-to-fine paradigm. Through further research, we find that the bag of proposals, either at the image level or the instance level, is established on discrete box sampling. This leads the pseudo box estimation into a sub-optimal solution, resulting in the truncation of object boundaries or the excessive inclusion of background. Hence, we conduct a series exploration of discrete-to-continuous optimization, yielding P2BNet++ and Point-to-Mask Network (P2MNet). P2BNet++ conducts an approximately continuous proposal sampling strategy by better utilizing spatial clues. P2MNet further introduces low-level image information to assist in pixel prediction, and a boundary self-prediction is designed to relieve the limitation of the estimated boxes. Benefiting from the continuous object-aware \textbf{\textit{pixel-level perception}}, P2MNet can generate more precise bounding boxes and generalize to segmentation tasks. Our method largely surpasses the previous methods in terms of the mean average precision on COCO, VOC, SBD, and Cityscapes, demonstrating great potential to bridge the performance gap compared with fully supervised tasks.