Editing Function of a real variable (section)

== Cardinality of sets of functions of a real variable ==
The [[cardinality]] of the set of real-valued functions of a real variable, <math>\mathbb{R}^\mathbb{R}=\{f:\mathbb{R}\to \mathbb{R}\}</math>, is <math>\beth_2=2^\mathfrak{c}</math>, which is strictly larger than the cardinality of the [[Continuum (set theory)|continuum]] (i.e., set of all real numbers).  This fact is easily verified by cardinal arithmetic:

<math display="block">\mathrm{card}(\R^\R)=\mathrm{card}(\R)^{\mathrm{card}(\R)}=
\mathfrak{c}^\mathfrak{c}=(2^{\aleph_0})^\mathfrak{c}=2^{\aleph_0\cdot\mathfrak{c}}=2^\mathfrak{c}.
</math>

Furthermore, if <math>X</math> is a set such that <math>2\leq\mathrm{card}(X)\leq\mathfrak{c}</math>, then the cardinality of the set <math>X^\mathbb{R}=\{f:\mathbb{R}\to X\}</math> is also <math>2^\mathfrak{c}</math>, since

<math display="block">2^\mathfrak{c}=\mathrm{card}(2^\R)\leq\mathrm{card}(X^\R)\leq\mathrm{card}(\R^
\R)=2^\mathfrak{c}.</math>

However, the set of [[continuous function]]s <math>C^0(\mathbb{R})=\{f:\mathbb{R}\to\mathbb{R}:f\ \mathrm{continuous}\}</math> has a strictly smaller cardinality, the cardinality of the continuum, <math>\mathfrak{c}</math>.  This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.<ref>{{Cite book|title=Principles of Mathematical Analysis|last=Rudin|first=W.|publisher=McGraw-Hill|year=1976|isbn=0-07-054235X|location=New York|pages=98–99}}</ref>  Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable.  By cardinal arithmetic:

<math display="block">\mathrm{card}(C^0(\R))\leq\mathrm{card}(\R^\Q)=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=
2^{\aleph_0}=\mathfrak{c}.</math>

On the other hand, since there is a clear [[bijection]] between <math>\R</math> and the set of constant functions <math>\{f:\R\to\R: f(x)\equiv x_0\}</math>, which forms a subset of <math>C^0(\R)</math>, <math>\mathrm{card}(C^0(\R)) \geq \mathfrak{c}</math> must also hold.  Hence, <math>\mathrm{card}(C^0(\R)) = \mathfrak{c}</math>.