  In a financial market with a continuous price process and proportional
transaction costs we investigate the problem of utility maximization of
terminal wealth. We give sufficient conditions for the existence of a shadow
price process, i.e.~a least favorable frictionless market leading to the same
optimal strategy and utility as in the original market under transaction costs.
The crucial ingredients are the continuity of the price process and the
hypothesis of "no unbounded profit with bounded risk". A counter-example
reveals that these hypotheses cannot be relaxed.
