A new perspective on continuity

Continuity of the function $f: \mathbb R \to \mathbb R$ asks if a small change in the input value creates a small change in the output. Equivalently, a continuous function must take two neighbors (points that are close in distance) of the input space to neighbors in the output space. If neighbors are taken to wildly distant points by the function, this would be a discontinuity.

Real analysis employs the epsilon-delta definition of limits to formalize this idea. $f$ is continuous at $x=a$ if and only if $\forall \varepsilon>0, \exists \delta>0,$ \[|x-a|<\delta \implies |f(x)-f(a)|<\varepsilon\]

This codifies the idea above, but in the backwards direction. That is, continuity implies all neighboring points of the output to have come from some neighbors in the input. Continuity on the real line is exacting – numerical distance and existence conditions are its essence.

But our intuition of continuity is more general than that. What does it mean for points to be neighbors? Is distance the only way points can be near each other? In fact, continuity may be redefined using the fundamental ideas of topology, one that gives rise to the study of surface structures and also the more specific continuity of real functions.

Open sets

A set has elements. Loosely speaking, define open sets as a set $X$ such that all elements $x\in X$ has an open neighborhood around them completely inside $X$. \[\forall x \in X, \exists N, \ x \in N \subset X\] For example, open intervals $(a \in \mathbb R,b \in \mathbb R)$ have the property that any point of the interval has another open subinterval containing it (a neighborhood).

More formally, open intervals are members of a topology on a set $X$. A topology $\mathcal T$ is the collection of $X$’s subsets ($\mathcal T \subset \mathcal P(X)$) such that \[\phi, X \in \mathcal T\] \[\forall U_i \in \mathcal T, \ \bigcup_i U_i \in \mathcal T\] \[\forall U_i \in \mathcal T, \ \bigcap_i^N U_i \in \mathcal T \text{ where } N \in \mathbb N\]

Members $U \in \mathcal T$ are called open sets of $X$.

Continuity

Consider the function $f: X \to Y$. Define $f$ to be continuous if and only if \[\forall V \in \mathcal T_Y, \ f^{-1}(V) \in \mathcal T_X\] i.e., \[V \text{ is an open set in } Y \implies f^{-1}(V) \text{ is an open set in } X\]

The inverse notation $f^{-1}(V)$ (the preimage) is simply the set of inputs that has its output a member of $V$, or $f^{-1}(V) := \{x \mid f(x) \in V\}$.

To be continued…