In this work, we study the tradeoff between locality and minimum distance of locally repairable codes (LRCs) with local codes that have an arbitrary distance requirement. Our approach is different from previous work in that we allow the locality requirements to vary arbitrarily from node to node. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by a dimension upper bound that does not depend on the distance. Furthermore, our bounds are directly expressed in terms of the locality requirement rather than the locality profile, therefore exactly correspond to conventional problem formulations.