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The intermediate value theorem states that if $f:[\alpha ,\beta ]\to \mathbb{R}$ is continuous and $y$ lies between $f(\alpha )$ and $f(\beta )$, then some $c\in [\alpha ,\beta ]$ satisfies $f(c)=y$.

It is enough to prove the caseLetChoose $x_n\in E$ with $x_n\to c$. Continuity gives $f(c)\le y$. If $f(c)<y$, continuity would make $f(x)<y$ for some $x>c$, contradicting that $c$ is an upper bound of $E$. Thus $f(c)=y$. The case with reversed endpoint inequalities follows by replacing $f$ with $-f$.