I am reading the book "Differentiable Manifolds" by Brickell and Clark.
on page 54, it is written that:If we denote the set of differentiable functions $f:M \to \mathbb{R}$ on a manifold $M$ whose domains include a given point $m\in M$ by $\mathcal{F}(m)$, then $\mathcal{F}(m)$ does not form a vector space over the set of real number $\mathbb{R}$. And that is because $\mathcal{F}(m)$ does not contain a function $-f$ such that $f + (-f) = 0$ for all $f\in \mathcal{F}(m)$!
I'm confused a little bit. Because if a function $f$ is defined at a point $m\in M$, then $-f$ is also defined at $m$. Am i making a mistake?
Thanks for any help.