A group $G$ is called locally indicable if for any finitely generated subgroup $H \subset G$, there is a nontrivial homomorphism from $H$ to the real additive group $(\mathbb{R},+)$. Is the Thompson group $F$ locally indicable?
Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log_2$ of the (right) slope of $h$ at $a$. This map is a nontrivial homomorphism of $H$ into $\mathbb{Z}$. A more fancy (but essentialy the same) proof is this: by Thurston's stability theorem (W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347 352), every group of $C^1$ diffeomorphisms of the closed interval is locally indicable, and by Ghys and Sergiescu, $F$ is a subgroup of the group of $C^\infty$ diffeomorphisms of the interval $[0,1]$ (see E. Ghys and V. Sergiescu: Sur un groupe remarquable de diﬀeomorphisms du cercle, Comm. Math. Helv. 62 (1987) 185–239, an easy proof using diagram groups can be found in V.S. Guba and M. Sapir. Diagram groups are totally orderable. J. Pure Appl. Algebra, 205(1):48–73, 2006). 

