Let $f:I\rightarrow R$ be continuous at $y\in I$. Suppose $f(y)>m$ for some $m \in R$. Prove there exists $\delta >0$ such that $f(x)>m$ for all $x \in I$ with $|x−y|<\delta $.
Proof: Let $f:I \rightarrow R$ be continuous at $y \in I$. By definition of continuity, $\forall \varepsilon >0 \, \exists \delta > 0$ such that if $x\in I$ and $|x−y|<\delta $ then $|f(x)−f(y)|<\varepsilon$. Suppose $f(y)>m$ for some $m\in R$. And that's where I get stuck.