Trying to solve Exercise 2.7 b) of Chapter III of Hartshorne's Algebraic Geometry I got stucked about an apparent contradiction.
The exercise asks to prove that $H^1(S^1, \mathcal{R})=0$, where $S^1$ is the circle with its usual topology and $\mathcal{R}$ is the sheaf of continuous real-valued functions.
I thought to apply the Mayer-Vietoris sequence for the cohomology with support to the closed subsets of $S^1$:
$U:=\{(cos\theta, sen\theta): \theta \in [o,\pi]\},$
$V:=\{(cos\theta, sen\theta): \theta \in [\pi,2\pi]\},$
Their intersection is the pair of points $\{(1,0),(-1,0)\}$.
We observe that $\mathcal{R}$ is a flasque sheaf (i.e. if $U\subseteq V$ are open sets then the restriction map $\mathcal{R}(V) \rightarrow \mathcal{R}(U)$ is a surjection, thanks to Exercise 2.3 c) of the same Chapter, all the cohomology group of grade greater than $0$ computed on a flasque sheaf are trivial. This would solve the exercise by itself but just keep thinking in this direction for a moment...). And we recall that $H^0_Y(X,\mathcal{F})=\Gamma_Y(X,\mathcal{F})$ we get the (short) exact sequence:
$0 \rightarrow \Gamma_{U \cap V}(S^1, \mathcal{R}) \rightarrow \Gamma_U(S^1, \mathcal{R})\oplus\Gamma_V(S^1, \mathcal{R}) \rightarrow \Gamma(S^1, \mathcal{R}) \rightarrow 0.$
Taking a closer look to the groups involved I noticed that $\Gamma_{U \cap V}(S^1, \mathcal{R})\simeq 0$ because the germ of a continuous function on a point which is zero outside of that point must be zero also on the point, by continuity. For the other terms $\Gamma_U(S^1, \mathcal{R}) \simeq \Gamma_V(S^1, \mathcal{R}) \simeq \mathcal{C}([0,1])_{\partial=0}$ the group of continuous functions on $[0,1]$ which are zero on the boundary. And $\Gamma(S^1, \mathcal{R}) \simeq \mathcal{C}([0,1])_{per}$ the group of continuos functions on $[0,1]$ such that $f(0)=f(1)$.
By the sequence above we deduce that $\mathcal{C}([0,1])_{per} \simeq \mathcal{C}([0,1])_{\partial=0}\oplus \mathcal{C}([0,1])_{\partial=0}$. But really I don't see it (I tried to do something with Borsuk-Ulam theorem but the matter is I don't think it is true).
Can someone please explain where my error lies? Or, otherwise, how the groups as above are isomorphic?