# constant functions and continuity

###### Theorem.

Suppose $X$ is path connected and $D$ is a countable  discrete topological space. If $f\colon X\to D$ is continuous, then $f$ is a constant function.

###### Proof.

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

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

for some $x,y\in X$. Since $X$ is path connected there is a continuous path $\gamma\colon[0,1]\to X$ such that $\gamma(0)=x$ and $\gamma(1)=y$. Then $f\circ\gamma\colon[0,1]\to D$ is continuous. Since $D$ has the subspace topology of $\mathbbmss{R}$, this result (http://planetmath.org/ContinuityIsPreservedWhenCodomainIsExtended) implies that also $f\circ\gamma\colon[0,1]\to\mathbbmss{R}$ 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 ConstantFunctionsAndContinuity 2013-03-22 15:17:31 2013-03-22 15:17:31 mathcam (2727) mathcam (2727) 12 mathcam (2727) Theorem msc 03E20