Porosity and Derivatives

Porosity or in particular σ-porosity in Rⁿ is a method of combining the ideas of measure and Baire category into one different and yet stronger notion. As it turns out, these sets are "sponge-like", having "uniformly no volume". The Sierpinski carpet and classic 1/3 Cantor set are both examples of iconic porous sets. The topic is generally refered to as the study of generic properties, as we may in fact develop strong enough ideas to say that porous conditions imply the probability of events. After introducing the notions, we will look at applications of the topic by discussing the answers to two questions: Are derivatives typically continuous? Are continuous functions typically differentiable?