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Monocular Depth Estimation (MDE) is a fundamental problem in computer vision with numerous applications. Recently, LIDAR-supervised methods have achieved remarkable per-pixel depth accuracy in outdoor scenes. However, significant errors are typically found in the proximity of depth discontinuities, i.e., depth edges, which often hinder the performance of depth-dependent applications that are sensitive to such inaccuracies, e.g., novel view synthesis and augmented reality. Since direct supervision for the location of depth edges is typically unavailable in sparse LIDAR-based scenes, encouraging the MDE model to produce correct depth edges is not straightforward. To the best of our knowledge this paper is the first attempt to address the depth edges issue for LIDAR-supervised scenes. In this work we propose to learn to detect the location of depth edges from densely-supervised synthetic data, and use it to generate supervision for the depth edges in the MDE training. To quantitatively evaluate our approach, and due to the lack of depth edges GT in LIDAR-based scenes, we manually annotated subsets of the KITTI and the DDAD datasets with depth edges ground truth. We demonstrate significant gains in the accuracy of the depth edges with comparable per-pixel depth accuracy on several challenging datasets. Code and datasets are available at \url{https://github.com/liortalker/MindTheEdge}.