# Real Analysis/Topological Continuity

Definition Let ${\displaystyle A\subseteq \mathbb {R} }$. Also, let ${\displaystyle f:A\to \mathbb {R} }$. ${\displaystyle f(x)}$ is continuous at ${\displaystyle x=c}$ iff for every open subset ${\displaystyle V}$ of ${\displaystyle f(A)}$, ${\displaystyle U\subseteq f^{-1}(V)}$ is open in ${\displaystyle A}$.