Let $ X $ be a topological space. Let $ P $ be a (pre)sheaf of sets on $ X $. Let $ U\subset X $ be an open set and let $ s\in PU $ be a section of $ P $. For $ x\in X $ let $ s_x $ denote the germ of $ s $ at $ x $.
I was wondering if $ s_x = s_y $ for two points $ x,y\in U $ necessarily implies $ x = y $. I tried playing with the sheaf of continuous functions on $ \mathbb R $, but I was not able to come up with a counterexample.