Definition, examples and properties

1. Continuous functions

   Let $X$ and $Y$ be topological spaces. The function $f\colon X\to Y$ is continuous if for any open set $V$ in $Y$, we have that $f^{-1}(V)$ is open in $X$.

   

2. Properties
   • If the topology of $Y$ is given by a basis $\mathcal{B}$, then $f\colon X\to Y$ is continuous if and only if $f^{-1}(V)$ is open in $X$ for every $V\in \mathcal{B}$.
   • If the topology of $Y$ is given by a subbasis $\mathcal{S}$, then $f\colon X\to Y$ is continuous if and only if $f^{-1}(V)$ is open in $X$ for every $V\in \mathcal{S}$.
   • A function $f\colon \mathbb{R}\to \mathbb{R}$ is continuous if and only if for every $x_{0}\in \mathbb{R}$ and $\epsilon>0$, there is a $\delta>0$ such that $\vert x-x_{0}\vert <\delta$ implies $\vert f(x)-f(x_{0})\vert <\epsilon$.
3. • Let $f\colon \mathbb{R}\to \mathbb{R}$ be continuous, $x_{0}\in \mathbb{R}$ and $\epsilon>0$.
   • We have that $f^{-1}((f(x_{0})-\epsilon,f(x_{0})+\epsilon))$ is open in $\mathbb{R}$ and contains $x_{0}$. Hence there is $\delta>0$ such that $(x_{0}-\delta,x_{0}+\delta)\subseteq f^{-1}(f(x_{0})-\epsilon,f(x_{0})+\epsilon)$.

Examples

1. Continuous functions
   • Any continuous function $f\colon \mathbb{R}\to \mathbb{R}$ from calculus courses is continuous in this sense.
   • Let $\mathbb{R}_{l}$ be the set of real numbers with the Sorgenfrey topology. Then the identity function $1_{\mathbb{R}}\colon \mathbb{R}\to \mathbb{R}_{l}$ is not continuous.
   • On the other hand, the identity $1_{\mathbb{R}}\colon \mathbb{R}_{l}\to \mathbb{R}$ is continuous.

Equivalences of continuity

1. Teorema

   Let $X,Y$ be topological spaces and $f\colon X\to Y$. Then the following are equivalent:

   • $f$ is continuous.
   • For every $A\subseteq X$, one has $f(\overline{A})\subseteq \overline{f(A)}$.
   • For every closed set $C$ in $Y$, one has $f^{-1}(C)$ is closed in $X$.
   • For each $x\in X$ and each neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U)\subseteq V$.

2. When the last condition is satisfied at $x_{0}\in X$, we say that $f$ is continuous at $x_{0}$.

Exercises

1. Let $f\colon X\to Y$ continuous. If $A\subseteq X$ and $x$ is a limit point of $A$, is $f(x)$ a limit point of $f(A)$?
2. If the singleton $\{x_{0}\}$ is open in $X$, prove that every function $f\colon X\to Y$ is continuous at $x_{0}$. Is the converse true?
3. Prove that $f\colon X\to Y$ is continuous if and only if for every $A\subseteq Y$ we have that $f^{-1}(A^{\circ})\subseteq (f^{-1}(A))^{\circ}$.