Generalized logarithmic exponent of linear differential systems in the topology of uniform convergence

We show that the generalized logarithmic exponent of an n-dimensional homogeneous linear differential system with continuous bounded coefficients on the half-line treated as a function of its right-hand side on the set of all such systems equipped with the topology of uniform convergence belongs exactly to the second Baire class if n ≥ 2 and to the zero Baire class if n = 1.