Continuous automorphism of a lie group in kobayashi's book

Question

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and i dont understand a thing at page 14.

My question is why $A_\varphi$ is continuous?

$G$ is a subgroup of transformation of a differentiable manifold, $G^*$ normal subgroup of $G.$

I've open Chevalley's book, Theory of Lie groups at the page 128 as Kobayashi suggested but there is nothing useful in there.

Any ideas on proving this?

Answer 1

Read the remark on that page carefully and make attention to local canonical coordinate chosen and remember that the 1-parameter group has the unique form $\text{exp}(tX)$.