Suppose that $S(t)$ is a real-matrix-valued (dimension $n\times n$) locally Lipschitz function defined on the non-negative real line. Suppose also that $S(t)$ is always symmetric and positive definite.
Consider a real-vector-valued (dimension $n\times 1$) locally Lipschitz function $e(t)$ defined on the same domain.
Is the real-valued function: $$v(t):=\sqrt{e(t)^TS(t)e(t)}$$ locally Lipschitz on the whole non-negative real line?