I am reading The Rising Sea - Foundations of Algebraic Geometry written by Ravi Vakil, and I'm asking for help because there was a point that didn't touch to me. Maybe it's because I don't have that much knowledge about differential geometry and algebraic geometry: I stated to study algebraic geometry a couple of weeks ago.
For motivating example of sheaf/presheaf, writer suggest the sheaf of smooth function. Vakil let $\mathcal O(U)=\{f:U\to\mathbb R\mid f\mathrm{\,is \,smooth}\}$ be ring of smooth functions defined over open set $U\subset\mathbb R^n$. Then we can define equivalence relation $\sim$ over $O=\{(f,U)\mid p\in U, f\in\mathcal O(U)\}$ when $p\in \mathbb R^n$ fixed as $(f,U_f)\sim(g,U_g)$ if and only if there exists open $p\in W\subset U,V$ such that $f|_W=g|_W$. Then $\mathcal O_p=O/\sim$ is ring, and has unique maximal ideal $\mathfrak{m}_p=\{\overline{(f,U)}\mid f(p)=0\}$ so $\mathcal O_p$ is local.
In Rhetorical question for experts in exercise 2.1.B says:
What goes wrong if the sheaf of continuous functions is substituted for the sheaf of smooth functions? What goes wrong if you use the sheaf of $\mathcal C^1$ functions?
Above definition, I cannot see any error while replacing smoothness with continuity. Definition of $\mathcal O_p$, does not use differential an single second. But however, seeing the author raise these issues has made me think that there might be a problem.
What goes wrong? It would go wrong while handling cotangent space or else: because Vakil made a notice that "$\mathfrak m_p/\mathfrak m_p^2$ is $\mathcal O_p/\mathfrak m_p\cong\mathbb R$-vector space and it turns out to be naturally toe cotangent space to the differentiable or analytic manifold at $p$".
Any help would be graceful. Thank you.
