Limits and Continuity, An Introduction

Introduction

In the 17^th century several mathematicians developed the concepts of limits and continuity, primarily to foster the development of calculus. If f(x) gets closer and closer to q, as x gets close to p, then the limit of f, at p, is q. If f(p) = q then f is continuous at q. Intuitively, a continuous function can be graphed without lifting your pencil offf the paper, no gaps or jumps.

The "close to" criterion, which will be made rigorous as we move along, relies on the notion of distance. This makes sense in our universe of 3 dimensions, where distance is well defined.

During the 18^th and 19^th centuries 3 space was generalized to finite dimensional space, infinite euclidean space, metric spaces, and finally topological spaces. These abstract spaces have abstract definitions of limits and continuity, involving open sets, but when those definitions are applied to the real world, they produce the "close to" criterion described above.

Most of the theorems in this section apply to Rⁿ, and some are restricted to one dimensional space. I'll try to be clear as we go.

Since this topic deals with limits and continuity in real space, it is sometimes called "real analysis". In fact there are plenty of text books with that title. So - call it real analysis if you like; I call it limits and continuity, primarily so I don't feel "boxed in". After all, some of these theorems apply to complex variables, and some to infinite vector spaces. But yes, most of the theorems are specific to real space - hence real analysis.