How to Extend a Function to Be Continuous

Continuous functions on a closed subset of a normal topological space can be extended

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

Formal statement [edit]

If ${\displaystyle X}$ is a normal space and

$${\displaystyle f:A\to \mathbb {R} }$$

is a continuous map from a closed subset ${\displaystyle A}$ of ${\displaystyle X}$ into the real numbers ${\displaystyle \mathbb {R} }$ carrying the standard topology, then there exists a continuous extension of ${\displaystyle f}$ to ${\displaystyle X;}$ that is, there exists a map

$${\displaystyle F:X\to \mathbb {R} }$$

continuous on all of ${\displaystyle X}$ with ${\displaystyle F(a)=f(a)}$ for all ${\displaystyle a\in A.}$ Moreover, ${\displaystyle F}$ may be chosen such that

$${\displaystyle \sup\{|f(a)|:a\in A\}~=~\sup\{|F(x)|:x\in X\},}$$

that is, if ${\displaystyle f}$ is bounded then ${\displaystyle F}$ may be chosen to be bounded (with the same bound as ${\displaystyle f}$ ).

History [edit]

L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when ${\displaystyle X}$ is a finite-dimensional real vector space. Heinrich Tietze extended it to all metric spaces, and Pavel Urysohn proved the theorem as stated here, for normal topological spaces.^{[1] [2]}

Equivalent statements [edit]

This theorem is equivalent to Urysohn's lemma (which is also equivalent to the normality of the space) and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing ${\displaystyle \mathbb {R} }$ with ${\displaystyle \mathbb {R} ^{J}}$ for some indexing set ${\displaystyle J,}$ any retract of ${\displaystyle \mathbb {R} ^{J},}$ or any normal absolute retract whatsoever.

Variations [edit]

If ${\displaystyle X}$ is a metric space, ${\displaystyle A}$ a non-empty subset of ${\displaystyle X}$ and ${\displaystyle f:A\to \mathbb {R} }$ is a Lipschitz continuous function with Lipschitz constant ${\displaystyle K,}$ then ${\displaystyle f}$ can be extended to a Lipschitz continuous function ${\displaystyle F:X\to \mathbb {R} }$ with same constant ${\displaystyle K.}$ This theorem is also valid for Hölder continuous functions, that is, if ${\displaystyle f:A\to \mathbb {R} }$ is Hölder continuous function with constant less than or equal to ${\displaystyle 1,}$ then ${\displaystyle f}$ can be extended to a Hölder continuous function ${\displaystyle F:X\to \mathbb {R} }$ with the same constant.^[3]

Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:^[4] Let ${\displaystyle A}$ be a closed subset of a normal topological space ${\displaystyle X.}$ If ${\displaystyle f:X\to \mathbb {R} }$ is an upper semicontinuous function, ${\displaystyle g:X\to \mathbb {R} }$ a lower semicontinuous function, and ${\displaystyle h:A\to \mathbb {R} }$ a continuous function such that ${\displaystyle f(x)\leq g(x)}$ for each ${\displaystyle x\in X}$ and ${\displaystyle f(a)\leq h(a)\leq g(a)}$ for each ${\displaystyle a\in A}$ , then there is a continuous extension ${\displaystyle H:X\to \mathbb {R} }$ of ${\displaystyle h}$ such that ${\displaystyle f(x)\leq H(x)\leq g(x)}$ for each ${\displaystyle x\in X.}$ This theorem is also valid with some additional hypothesis if ${\displaystyle \mathbb {R} }$ is replaced by a general locally solid Riesz space.^[4]

See also [edit]

• Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
• Hahn–Banach theorem – Theorem on extension of bounded linear functionals
• Whitney extension theorem – Partial converse of Taylor's theorem

References [edit]

1. ^ "Urysohn-Brouwer lemma", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. ^ Urysohn, Paul (1925), "Über die Mächtigkeit der zusammenhängenden Mengen", Mathematische Annalen, 94 (1): 262–295, doi:10.1007/BF01208659, hdl:10338.dmlcz/101038 .
3. ^ McShane, E. J. (1 December 1934). "Extension of range of functions". Bulletin of the American Mathematical Society. 40 (12): 837–843. doi:10.1090/S0002-9904-1934-05978-0.
4. ^ ^{a b} Zafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF). Turkish Journal of Mathematics. 21 (4): 423–430.

• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN978-0-13-181629-9. OCLC 42683260.

External links [edit]

• Weisstein, Eric W. "Tietze's Extension Theorem." From MathWorld
• Mizar system proof: http://mizar.org/version/current/html/tietze.html#T23
• Bonan, Edmond (1971), "Relèvements-Prolongements à valeurs dans les espaces de Fréchet", Comptes Rendus de l'Académie des Sciences, Série I, 272: 714–717 .

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Source: https://en.wikipedia.org/wiki/Tietze_extension_theorem