----- > [!proposition] Proposition. ([[continuity condition for bounds to be preserved under closure]]) > Let $X$ be a [[topological space]], $A \subset X$, and $h:X \to \mathbb{R}$ a [[continuous]] function. Suppose that there is a constant $c$ s.t. $h(x)\leq c$ for all $x \in A$. Prove that $h(x) \leq c$ for all $x \in \overline{A}$. > [!proof]- Proof. ([[continuity condition for bounds to be preserved under closure]]) > ~ > It suffices to show the result only for any $x$ that is a [[limit point]] of $A$. > Suppose $h(x)>c$. Then since the set $\{ y \in \mathbb{R}: y >c \}$ is open in $\mathbb{R}$, there exists a [[neighborhood]] $U\ni$ $h(x)$ such that $y > c$ for all $y \in U$. Now, $h^{-1}(U)$ is an open [[neighborhood]] of $x$ in $X$ (by [[continuous|continuity]]), and thus intersects $A$ at some point $x'$ other than $x$ itself. But then we have simultaneously that $h(x') \leq c$ (since $x' \in A$) and $h(x')>c$ (since $h(x') \in U$). This is a contradiction. So we must have $h(x) \leq c$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```