I think $f(t) = e^{X(t)}$ is continuous over $t \in \mathbb{R}$ when the complex matrix valued mapping $t\mapsto X(t)$ is continuous - regardless of whether $X$ is infinite or finite dimension. I tried to think of a proof, but without much success.

The hard part is the infinite-dimensional case, and I do not know what a proof could look like.

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    $\begingroup$ The composed of two continuous maps is continuous. $\endgroup$ – Gustave Mar 10 at 11:51
  • $\begingroup$ @Gustave: So you need to show $A \mapsto e^A$ is continuous. For marix $A$ or bounded operator $A$ it is clear. But the OP does not spell out his/her assumptions. Physicists talk about semigroups of unbounded operators, for example. $\endgroup$ – GEdgar Mar 10 at 12:34
  • $\begingroup$ @GEdgar A semigroup by definition is continuous, we don't talk here about the $X(t)$ as operator but as one variable function. Cordially. $\endgroup$ – Gustave Mar 10 at 18:18

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