Initial states problem with ODE

Question

I have the following problem: What can we say about the funcions $t_0 \rightarrow y(t,t_0,y_0)$ and $ y_0 \rightarrow y(t,t_0,y_0)$ where $y(t,t_0,x_0)$ is the solution of the following Caychy problem $$y'(t)=f(t,y(t)), \\ y(t_0)=y_0$$ with $f$ is gloally lipcshitz. Are they are differentiabl! or invertible ? Thank you.