I am trying to formalize the following proof on Perko's Differential Equations and Dynamical Systems, which says that a periodic orbit has index +1.
My only problem is trying to prove that the map $g$ is continuous on $T$. Geometrically it seems obvious, and I am trying to prove it using limits, but I do not get anything. The following figure (also from Perso's book) is a nice illustration:
All follows from the definition of the vector field $\bf u$. For example, on the line $s=t$ you have $$ \lim_{t\to s^+}\frac{x(t)-x(s)}{\|x(t)-x(s)\|}=\lim_{t\to s^+}\frac{\frac{x(t)-x(s)}{t-s}}{\left\|\frac{x(t)-x(s)}{t-s}\right\|}=\frac{x'(s)}{\|x'(s)\|}={\bf u}(x(s)). $$
The proof and figures are copied from Coddington and Levinson's book, really the best book there is on this particular topic.