Following Andreas Blass's hints, one possibility to show the continuity is the following. Let $d:=n-k$ be the dimension of the cutting affine spaces.
First, let's say that a face of a polytope is a bad face if it spans an affine space containing a $d$ dimensional subspace parallel to the cutting space; otherwise, call it a nice face. If all faces of a polytope are nice, let's say it's a nice polytope, after all.
For a nice simplex, one checks that the function $f$ is continuous on the whole $a$-space.
By genericity, any polytope can be subdivided into nice simplexes, plus a number of simplexes whose bad faces are included into some bad faces of the polytope.
In other words, a generic triangulation produces a simplicial complex where no bad faces are introduced. This shows that the points of discontinuity of the function $f$ of the polytope are exactly the projection in the $a$-space of its bad faces.
