The sensitivity of image classifiers to small perturbations in the input is often viewed as a defect of their construction. We demonstrate that this sensitivity is a fundamental property of classifiers. For any arbitrary classifier over the set of $n$-by-$n$ images, we show that for all but one class it is possible to change the classification of all but a tiny fraction of the images in that class with a perturbation of size $O(n^{1/\max{(p,1)}})$ when measured in any $p$-norm for $p \geq 0$. We then discuss how this phenomenon relates to human visual perception and the potential implications for the design considerations of computer vision systems.