This may be a trivial question. If so, apologies in advance.
Let a $1$-form $\omega$ be given. Suppose that for any line segments $C_1=[i,j]$, $C_2=[j,k]$, $C_3=[i,k]$, we have that $\int_{C_1} \omega + \int_{C_2} \omega = \int_{C_3} \omega$.
My question is whether this implies that $\omega$ is exact. If this were true for any curves $C_1,C_2,C_3$, I believe the answer would be affirmative. I want to make sure that restricting attention to finitely (or countably) many line segments is without loss of generality.