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 we can assume that $D$ is either $\{1,\ldots, n\}$ , $n\ge 2$ or , and these are equipped with the subspace topology of . Suppose $f(X)$ has at least two distinct elements, say 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 , this result implies that also 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.