On decision regions of narrow deep neural networks

We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well as for the ReLU activation function. This complements recent results on approximation capabilities by Hanin and Sellke (2017) and connectivity of decision regions by Nguyen et al. (2018) for such narrow neural networks. Our results are illustrated by means of numerical experiments.