When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map $$ E\times E^*\to\mathbb R,\qquad (x,L)\mapsto L(x) $$ is not continuous with respect to any locally convex topology on $E^*$, where $E$ is a non normable locally convex space and $E^*$ is the set of all continuous linear functionals $E\to\mathbb R$. Then they argue that, for this reason, it is not good to define $f\colon E\to\mathbb R$ to be continuously differentiable by requiring $$ x\mapsto Df(x)\in E^*$$ to be continuous with respect to some locally convex topology on $E^*$ (say, for example, the finest locally convex topology, in order to make the strongest assumption).

My question is: what's the problem with this definition? More precisely, what is an example of missing properties of $f\in C^1$ defined as above? I was thinking about continuity of $f$ being not implied by $f\in C^1$, or maybe the failure of the chain rule, but I didn't find an explicit issue. For example, the definition is strong enough to have a mean value theorem $$ f(x+h)-f(x)=\int_0^1 Df(x+th)h\,dt$$

Moreover, which of these classical properties actually require the evaluation map to be continuous?

I am particularly interested in the case of $E$ Fréchet space.

[cross-posted from MSE]