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 $ℝ$. 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 $ℝ$, this result (http://planetmath.org/ContinuityIsPreservedWhenCodomainIsExtended) implies that also $f\circ \gamma :[0,1]\to ℝ$ 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