Consider the following problem: $\dot{x} = A(t) x$. Let $X_{A}(t,s)$ be the Cauchy operator of the following system. Let $M$ be the space of continuous functional matrices. Let's denote by: $$ \rho(A, B) = \sup_{t \in \mathbb{R}_{+}} \|A(t) - B(t)\|, \text{ where } A,B \in M $$ Let's consider the upper-limit function $\phi(A) = \underset{{t-s \to +\infty}}{\overline{\lim}} \dfrac{1}{t - s}\log\|X_A(t,s)\|$. We want to show that $\forall a \in \mathbb{R}_{+}$ the set: $\{A\in M: \phi(A) < a\}$(i.e. Lyapunov's exponent is bounded with $a$) is open w.r.t. the metrics defined above.
I'was trying to use the simple definition of the openness in metric space (assume $a$ is fixed): $$ B_{\epsilon} (A) \subset A, \text{ for } A \in M $$ Which is equal to: $\exists B : \phi(A), \phi(B) < r$ and $\rho(A,B) < \epsilon$.
We can try to denote: $\{\phi(A) <a \} \sim \{A: \|X_A\| \sim e^{ r(t-s) + o(t-s)}\}$, but here I guess we need to use some more fundamental fact about $X_A$ (maybe continuous dependency on $A$).
UPD. We can consider the following problem as following. There is exists $Q(t) \in M$: $B(t) = A(t) + Q(t)$, $\rho(A,B) = \sup_t \|Q(t)\| < \epsilon$ and $\phi(A), \phi(B) < a$.
I thought about diagonal operator with small enough values on the diagonal (since I guess we can only consider the diagonal matrices).
Any hints?
