Editing Tietze extension theorem (section)

==Variations==
If <math>X</math> is a metric space, <math>A</math> a non-empty subset of <math>X</math> and <math>f : A \to \R</math> is a [[Lipschitz continuous]] function with Lipschitz constant <math>K,</math> then <math>f</math> can be extended to a Lipschitz continuous function <math>F : X \to \R</math> with same constant <math>K.</math>
This theorem is also valid for [[Hölder condition|Hölder continuous functions]], that is, if <math>f : A \to \R</math> is Hölder continuous function with constant less than or equal to <math>1,</math> then <math>f</math> can be extended to a Hölder continuous function <math>F : X \to \R</math> with the same constant.<ref>{{cite journal|last1=McShane|first1=E. J.|title=Extension of range of functions|journal=Bulletin of the American Mathematical Society|date=1 December 1934|volume=40|issue=12|pages=837–843|doi=10.1090/S0002-9904-1934-05978-0|doi-access=free}}</ref>

Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:<ref name="Zaf:97">{{cite journal|last1=Zafer|first1=Ercan|title=Extension and Separation of Vector Valued Functions|journal=Turkish Journal of Mathematics|date=1997|volume=21|issue=4|pages=423–430|url=http://journals.tubitak.gov.tr/math/issues/mat-97-21-4/mat-21-4-4-e2104-04.pdf}}</ref>
Let <math>A</math> be a closed subset of a normal topological space <math>X.</math> If <math>f : X \to \R</math> is an [[upper semicontinuous]] function, <math>g : X \to \R</math> a [[lower semicontinuous]] function, and <math>h : A \to \R</math> a continuous function such that <math>f(x) \leq g(x)</math> for each <math>x \in X</math> and <math>f(a) \leq h(a) \leq g(a)</math> for each <math>a \in A</math>, then there is a continuous
extension <math>H : X \to \R</math> of <math>h</math> such that <math>f(x) \leq H(x) \leq g(x)</math> for each <math>x \in X.</math> 
This theorem is also valid with some additional hypothesis if <math>\R</math> is replaced by a general locally solid [[Riesz space]].<ref name="Zaf:97" />

Dugundji (1951) extends the theorem as follows: If <math>X</math> is a metric space, <math>Y</math> is a [[locally convex topological vector space]], <math>A</math> is a closed subset of <math>X</math> and <math>f:A\to Y</math> is continuous, then it could be extended to a continuous function <math>\tilde f</math> defined on all of <math>X</math>. Moreover, the extension could be chosen such that <math>\tilde f(X)\subseteq \text{conv} f(A)</math>