Overview of Limits, Continuity, and Uniform Continuity
1. Limits Defined Abstractly
We will assume for the moment that we have some notion of neighborhoods of points \(x\) in some suitable set. Abstractly, a collection of sets will be called a system of compatible neighborhoods of \(x\) when they all contain the point \(x\) and are closed under arbitrary unions and finite intersections.
Definition
Assuming that suitable definitions of neighborhods have been made, given a function \(f \ : \ X \rightarrow Y\), we say that
\[ \lim_{x \rightarrow p} f(x) = L \]
when for every neighborhood \(N_Y \subset Y\) of the point \(L\), there is a neighborhood \(N_X \subset X\) of \(p\) such that \(f\) maps \(N_X \setminus \{p\}\) into \(N_Y\), i.e., every \(x \in N_X \setminus \{p\}\) has \(f(x) \in N_Y\).
We say that \(f\) is continuous at \(p\) when \(\lim_{x \rightarrow p} f(x) = f(p)\).
Example
Within the set \(\mathbb N \cup \{\infty\}\), let us define a neighborhood of infinity to be any set of the form \(U_N := \{ n \ : \ n > N\} \cup \{\infty\}\) for some \(N\). Arbitrary unions of such sets must have the same form (simply because all subsets of the natural numbers have a smallest element and \(\bigcup_{N \in E} U_N = U_{\min E}\)). Within \({\mathbb R}\), let us define a neighborhood of \(x\) to be any open set containing \(x\). Then for a function \(f : {\mathbb N} \cup \{\infty\} \rightarrow {\mathbb R}\), the definition just made for the statement \(\lim_{n \rightarrow \infty} f(n) = L\) coincides with the sequential definition of this same limit.
2. Uniform Continuity
In this unit, we will encounter another sort of generalization of continuity known as uniform continuity. In the case of real functions on an interval \(I\), we have the familiar \(\epsilon\)-\(\delta\) definition of continuity at every point \(x \in I\):
Definition (Continuity)
\[{}\forall x \in I, \ \forall \epsilon > 0, \ \exists \delta > 0,{}\]
\[{}\ \forall y \in I, \ |x-y| < \delta{}\]
\[{}\Rightarrow |f(y) - f(x)| < \epsilon{}\]
Observe that the order of quantifiers allows \(\delta\) to depend on the choice of \(x\) and \(\epsilon\), but not on the choice of \(y\). Uniform continuity is a stronger notion of continuity which requires that one be able to find \(\delta\) which may depend on \(\epsilon\) but not on either \(x\) or \(y\):
Definition (Uniform Continuity)
\[{}\forall \epsilon > 0, \ \exists \delta > 0, \ \forall x \in I,{}\]
\[{}\ \forall y \in I, \ |x-y| < \delta{}\]
\[{}\Rightarrow |f(y) - f(x)| < \epsilon{}\]
All uniformly continuous functions are continuous, but not vice-versa.
Below is a graph of a function on \((0,\infty)\) which is continuous but not uniformly continuous. The key feature to observe is that the function oscillates roughly like \(\sin x\) except the oscillations become more and more rapid as \(x \rightarrow \infty\). If you fix \(\epsilon = 1\) and any potential value of \(\delta > 0\), it is always possible by moving far enough to the right to find two points \(x,y \in (0,\infty)\) such that \(|x-y| < \delta\) but \(|f(x) - f(y)| > \epsilon\) (by, for example, taking \(x\) to be at the crest of one oscillation and \(y\) to be at the adjacent trough).
