Antimicrobial efficacy is traditionally described by a single value, the minimal inhibitory concentration (MIC), which is the lowest concentration that prevents visible growth of the bacterial population. As a consequence, bacteria are classically qualitatively categorized as resistant if therapeutic concentrations are below MIC and susceptible otherwise. However, there is a continuity in the space of the bacterial resistance levels. Here, we introduce a model of within-host evolution of resistance under treatment that considers resistance as a continuous quantitative trait, describing the level of resistance of the bacterial population. The use of integro-differential equations allows to simultaneously track the dynamics of the bacterial population size and the evolution of its level of resistance. We analyze this model to characterize the conditions; in terms of (a) the efficiency of the drug measured by the antimicrobial activity relatively to the host immune response, and (b) the cost-benefit of resistance; that (i) prevents bacterial growth to make the patient healthy, and (ii) ensures the emergence of a bacterial population with a minimal level of resistance in case of treatment failure. We investigate how chemotherapy (i.e., drug treatment) impacts bacterial population structure at equilibrium, focusing on the level of evolved resistance by the bacterial population in presence of antimicrobial pressure. We show that this level is explained by the reproduction number R0. We also explore the impact of the initial bacterial population size and their average resistance level on the minimal duration of drug administration in preventing bacterial growth and the emergence of resistant bacterial population.