Microscopic random quenched impurities may or may not produce rounding of a first-order phase transition. We derive a criterion for the appearance of rounding due to local fluctuations in thermodynamic phase. Such fluctuations occur when the free-energy lowering due to taking advantage of local fluctuations in impurity density more than offsets the free-energy cost of the interface produced. The argument also predicts the spatial scale of such phase fluctuations, when they occur. In some situations this scale is just the coherence length ξ in others, the inhomogeneity develops over "domains," which may be much larger than ξ. Near a second-order transition our criterion reduces to the one due to Harris. We specifically discuss what happens when a first-order transition becomes second order as an external parameter is varied.