Localizing functional regions of objects or affordances is an important aspect of scene understanding and relevant for many robotics applications. In this work, we introduce a pixel-wise annotated affordance dataset of 3090 images containing 9916 object instances. Since parts of an object can have multiple affordances, we address this by a convolutional neural network for multilabel affordance segmentation. We also propose an approach to train the network from very few keypoint annotations. Our approach achieves a higher affordance detection accuracy than other weakly supervised methods that also rely on keypoint annotations or image annotations as weak supervision.