Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows.

Equip it with the locally convex topology of the colimit. Specifically, it is given the finest locally convex topology so that all of the inclusions of finite summands are continuous. What is important is that a linear functions $\phi \colon \sum_{\mathbb{R}} \mathbb{R} \to \mathbb{R}$ is continuous if and only if its restriction to any finite summand is continuous. Thus

**all**linear functionals $\sum_{\mathbb{R}} \mathbb{R} \to \mathbb{R}$ are continuous. Technical note: with this topology, it is complete.Now define the family of smooth curves to be those curves $c \colon \mathbb{R} \to \sum_{\mathbb{R}} \mathbb{R}$ with the property that $\psi \circ c \colon \mathbb{R} \to \mathbb{R}$ is smooth for all $\psi \in \prod_{\mathbb{R}} \mathbb{R}$ (that is, for all

**continuous, linear**maps $\psi : \sum_{\mathbb{R}} \mathbb{R} \to \mathbb{R}$). Note that curves are not assumed to be continuous (though they will be).Now define the family of smooth functions to be those $f \colon \sum_{\mathbb{R}} \mathbb{R} \to \mathbb{R}$ for which $f \circ c \colon \mathbb{R} \to \mathbb{R}$ is smooth for all smooth curves $c$. Again, smooth functions are not assumed to be continuous.

Here's the question: are all smooth functions continuous?

If we took a countable sum then this would be true. For the uncountable product, Kriegl and Michor show (Example 4.8, pp37-38 of *A Convenient Setting of Global Analysis*) that this is not true. So I suspect the answer to be false, but don't know if this is known or not.

cancheck on smooth curves (a paper in Math Scand, no less!) - that's the starting point of the whole Frolicher machine. But you're right for $C^1$. Turns out that if you impose the condition that the derivatives be Lipschitz, then you can check on the corresponding class of curves (details in the early chapters of Kriegl and Michor's book). – Loop Space Nov 13 '09 at 10:32