The right homotopy shift in the fundamental groups of inverse limits

We introduce the right homotopy shift of paths and loops in the inverse limit of a single upper semi-continuous multivalued function on the unit interval. Consequently we obtain a shift in the fundamental group, which turns out to be an injective map under conditions usually assumed for such one-dimensional limits. As a result we obtain strong restriction on the fundamental groups of such inverse limits: they are not finitely generated and are often trivial or uncountable.