# constant functions and continuity

It is easy to see that every constant function between topological
spaces^{} is continuous^{}. A converse^{} result is as follows.

###### Theorem.

Suppose $X$ is path connected and $D$ is a countable^{}
discrete topological space. If $f\mathrm{:}X\mathrm{\to}D$ is continuous,
then $f$ is a constant function.

###### Proof.

By this result (http://planetmath.org/FiniteAndCountableDiscreteSpaces) we can assume that $D$ is either $\{1,\mathrm{\dots},n\}$, $n\ge 2$ or $\mathbb{Z}$, and these are equipped with the subspace topology of $\mathbb{R}$. Suppose $f(X)$ has at least two distinct elements, say $\alpha ,\beta \in \mathbb{Z}$ so that

$$f(x)=\alpha ,f(y)=\beta $$ |

for some $x,y\in X$. Since $X$ is path connected there is a continuous path
$\gamma :[0,1]\to X$ such that $\gamma (0)=x$ and $\gamma (1)=y$.
Then $f\circ \gamma :[0,1]\to D$ is continuous.
Since $D$ has the subspace topology of $\mathbb{R}$,
this result (http://planetmath.org/ContinuityIsPreservedWhenCodomainIsExtended)
implies that
also $f\circ \gamma :[0,1]\to \mathbb{R}$ is continuous.
Since $f\circ \gamma $
achieves two different values, it achieves uncountably many values,
by the intermediate value theorem.
This is a contradiction^{} since $f\circ \gamma ([0,1])$
is countable.
∎

Title | constant functions and continuity |
---|---|

Canonical name | ConstantFunctionsAndContinuity |

Date of creation | 2013-03-22 15:17:31 |

Last modified on | 2013-03-22 15:17:31 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 12 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 03E20 |