A. Given a totally (linearly) ordered set $A$ and a not constant function $f: A \rightarrow B$, are there any known conditions which guarantee there is no order automorphism $g :A \rightarrow A$, $ g\neq Id $ such that $f=f \circ g$?
B. Does it make any difference to the question $A$ above if the set $A$ is the Real numbers?
C. Does it make any difference to the question $A$ above if the set $A$ is a closed interval of the real numbers?
D. Does it make any difference to the question $A$ above if the set $A$ is equipped with the corresponding order topology, $B$ is a topological space and $f$ is continuous?
E. Is such a condition that $ f $ does not take at least two different values at least countably infinite times each?