I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance generating its topology, and endow the set of self-homeomorphisms of $S$ with the distance max(uniform distance between two maps, uniform distance between their inverses)).

*For any $\varepsilon>0$ there exists $\eta>0$ such that for any self-homeomorphism $f$ of $S$ whose distance to the identity is $<\eta$, there is an isotopy $t\mapsto f_t$ from $f$ to the identity that stays at distance $<\varepsilon$ to the identity.*

I'd be interested in a reference and/or a hint of the proof.

Note: Unless I'm too tired and got it wrong, this result implies (and is implied by the fact) that the group of self-homeomorphisms of $S$ is locally path-connected (in the sense that every point has a path-connected neighbourhood, not necessarily open), whence the title of this question.