Single Peakedness and Giffen Demand

I provide a simple example of a single peaked utility function that generates a Giffen demand. The utility function is smooth, non piecewise defined, strictly concave but not globally increasing. A full characterization of the parameter conditions under which the Giffen demand arises is provided. In addition the properties of the demand function are studied: I find that the inferior commodity with a Giffen demand must be cheaper relatively to a substitue and that Giffen demand arises at relatively low levels of income. However it is not required that the share of income spent on that commodity be large.