It is well known that the existence and uniqueness of Cournot equilibrium would extend to environments where firms prefer to be not active. However, we show that differentiated Bertrand oligopolies with constant unit costs and continuous best replies do not need to satisfy supermodularity (Topkis (1979)) or single crossing property (Milgrom and Shannon (1994)). Moreover, best replies might be negatively sloped and there are infinitely many undominated Bertrand-Nash equilibria on a wide range of parameter values when the number of firms is more than two. These results are very different from the existing literature on Bertrand models, where uniqueness, supermodularity, and the single crossing property usually hold under a linear market demand assumption and best reply functions slope upwards. We fully characterize the set of undominated equilibria. We provide an iterative algorithm to find the set of players that are active in any equilibrium, and show that this set is the same in all undominated equilibria.