Continuity In Banach Spaces

The main theme of this document and much of the author's research so far is to use porosity (as well as category) to describe how typical a variety of continuity conditions are within certain collections of functions. The intuition behind the work is that ``most things should behave according to a pattern or principle of action''. The pattern that the author has in mind is that things should: flow, be localized, and oscillate. Indeed, this theme occurs throughout the authors included work. As a side note: the reader with a very particular background and side interest may recognize these three principles as Yu-Won-Hwa, an observation that will help with understanding future naming conventions. In Chapter 1, we study what properties a typical bounded real valued derivative possesses, in terms of continuity. We first prove results for finite dimensional domains. Additionally, we obtain some results when the domain is a subset of a general Banach space with a Frechet differentiable norm. In Chapter 2, we study porosity in relation to bounded variation. In particular, we show that when we suitably norm the space of functions with bounded variation, then the Cantor function becomes the typical example of a function in that space. In Chapter 3, we study how typical (in the sense of both category and porosity) it is for a function that is twice partial differentiable to have equal mixed partial derivatives. As it turns out, the ability to satisfy Clairaut's Theorem is infrequent. In Chapter 4, we study a general condition that implies we have an open, dense, co-porous set whenever we are looking at a set defined by a seminorm in a particular way. This allows us to prove a number of unifying results (without having to individually go through many cases involving much calculation). In Chapter 5, we introduce a few open questions that the author has recently been working on directly, or has formulated for further study. In the Appendix, we introduce the concept of porosity in an easy to follow format, with illustrative diagrams to guide the reader in their pursuit of intuition