Existence, continuity and utility representation of strictly monotonic preferences on continuum of goods commodity spaces

It is an easy task for most commodity spaces, to find examples of strictly monotonic preference relations. For example, in the space of bounded sequences of real numbers. However, it is not easy for spaces like the space of bounded functions defined in the real interval [0, 1]. In this note we investigate the roots of this difficulty. We show that strictly monotonic preferences on the space of bounded function on any set K always exist. However, if K is uncountable no such preference is continuous and none of them have a utility representation.