I am seeing the definition of flow in a metric space:
$f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$.
Note that the condition does not require $f(x,0)=x$. My question is this: If $f_t(x):=f(x,t)$ is a homeomorphism for each t, Then we can conclude that $f(x,0)=x$ for all x?
If one can suggest examples of flows under this definition I will be very grateful.