Smooth solutions to systems of linear inequalities on $\mathbb{R}$

We study the one-dimensional case of the problem of deciding when a system of linear inequalities $$\sum _{j=1}^M{A}_{ij}(x)F_j(x)\le f_i(x),i=1,\dots ,N,$$ with fixed semialgebraic coefficients $A_{ij}\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$, admits a solution $F=(F_1,\dots ,F_M)\in C^m(\mathbb{R},{\mathbb{R}}^M)$. The analogous problem for systems of linear equations has a finite linear-differential criterion. For $x\in {\mathbb{R}}^{n\ge 2}$, the corresponding finite linear-differential-inequality criterion is known to fail. The obstruction comes from the fact that there are infinitely many directions to approach a point, and the fact that an infinite intersection of polytopes need not remain a polytope. The purpose of this paper is to prove that this obstruction disappears in dimension one. More precisely, solvability is characterized by finitely many linear ordinary differential inequalities in the data $f=(f_1,\dots ,f_N)$ with semialgebraic coefficients.