Abstract
In this paper, we study the questions of the discontinuity of the extreme oscillation exponents on the set of linear homogeneous differential systems with continuous coefficients on the positive axis. The existence of points on a set of differential systems is established in which all the higher and lower exponents of the oscillation zeros, roots, and hypercorns are not only not continuous, but are not continuous either from above or from below. Moreover, the non-invariance of the extreme exponent of oscillations with respect to infinitesimal perturbations has been proved. When proving the results of this work, the cases of parity and odd order of the matrix of the system are considered separately.
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