# supremum over closure

###### Theorem 1.

Let $f\mathrm{:}\mathrm{R}\mathrm{\to}\mathrm{R}$ be a continuous function^{} and $A\mathrm{\subseteq}\mathrm{R}$. Then $\underset{x\mathrm{\in}A}{\mathrm{sup}}\mathit{}f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}\underset{x\mathrm{\in}\overline{A}}{\mathrm{sup}}\mathit{}f\mathit{}\mathrm{(}x\mathrm{)}$, where $\overline{A}$ denotes the closure^{} of $A$.

###### Proof.

The theorem is clearly true for $A=\mathrm{\varnothing}$. Thus, it will be assumed that $A\ne \mathrm{\varnothing}$.

Since $A\subseteq \overline{A}$, we have $\underset{x\in A}{sup}f(x)\le \underset{x\in \overline{A}}{sup}f(x)$.

Suppose first that $\underset{x\in \overline{A}}{sup}f(x)=\mathrm{\infty}$. Let $r\in \mathbb{R}$. Then there exists ${x}_{0}\in \overline{A}$ with $f({x}_{0})\ge r+1$. Since $f$ is continuous, there exists $\delta >0$ such that, for any $x\in \mathbb{R}$ with $$, we have $$. Since ${x}_{0}\in \overline{A}$, there exists ${x}_{1}\in A$ with $$. (Recall that $x\in \overline{A}$ if and only if every neighborhood^{} of $x$ intersects $A$.) Thus, $f({x}_{1})-f({x}_{0})>-1$. Therefore, $f({x}_{1})>f({x}_{0})-1\ge r+1-1=r$. Hence, $\underset{x\in A}{sup}f(x)=\mathrm{\infty}$.

Now suppose that $\underset{x\in \overline{A}}{sup}f(x)=R$ for some $R\in \mathbb{R}$. Let $\epsilon >0$. Then there exists ${x}_{2}\in \overline{A}$ with $f({x}_{2})\ge R-\frac{\epsilon}{2}$. Since $f$ is continuous, there exists ${\delta}^{\prime}>0$ such that, for any $x\in \mathbb{R}$ with $$, we have $$. Since ${x}_{2}\in \overline{A}$, there exists ${x}_{3}\in A$ with $$. Thus, $f({x}_{3})-f({x}_{2})>\frac{-\epsilon}{2}$. Therefore, $f({x}_{3})>f({x}_{2})-\frac{\epsilon}{2}\ge R-\frac{\epsilon}{2}-\frac{\epsilon}{2}=R-\epsilon $. Hence, $\underset{x\in A}{sup}f(x)\ge R$.

In either case, it follows that $\underset{x\in A}{sup}f(x)=\underset{x\in \overline{A}}{sup}f(x)$. ∎

Note that this theorem also holds for continuous functions $f:X\to \mathbb{R}$, where $X$ is an arbitrary topological space^{}. To prove this fact, one would need to slightly adjust the proof supplied here.

Title | supremum over closure |
---|---|

Canonical name | SupremumOverClosure |

Date of creation | 2013-03-22 17:08:22 |

Last modified on | 2013-03-22 17:08:22 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 9 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 06A05 |

Classification | msc 26A15 |