Nowhere constant families of maps and resolvability

If $X$ is a topological space and $Y$ is any set then we call a family $\mathcal{F}$ of maps from $X$ to $Y$ nowhere constant if for every non-empty open set $U$ in $X$ there is $f \in \mathcal{F}$ with $|f[U]| > 1$, i.e. $f$ is not constant on $U$. We prove the following result that improves several earlier results in the literature. If $X$ is a topological space for which $C(X)$, the family of all continuous maps of $X$ to $\mathbb{R}$, is nowhere constant and $X$ has a $\pi$-base consisting of connected sets then $X$ is $\mathfrak{c}$-resolvable.
Comment: 6 pages