Continuity Test for Real-Valued Functions/Everywhere Continuous

Theorem
Let $\struct{S, \tau}$ be a topological space.

Let $f: S \to \R$ be a real-valued function.

Then $f$ is everywhere continuous :
 * $\forall x \in S : \forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$

Proof
By definition, $f$ is everywhere continuous $f$ is continuous at every point $x \in S$.

From Continuity Test for Real-Valued Functions, $f$ is continuous at every point $x \in S$ :
 * $\forall x \in S : \forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$