Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$ to be the stalk at $0$ of the sheaf of continuous $\mathbb{R}^n$ valued functions on $\mathbb{R}^m$. I'm trying to get a better idea of the topology on this space of germs. So, here's my question: Are there any continuous functions $F\colon C_0^{m,n} \to \mathbb{R}$ other than those that factor through the evaluation map $[f] \mapsto f(0)$?

If there are more, what are some examples? If there aren't, what's the underlying topological reason?