Multidrug resistance consists of a series of genetic and epigenetic alternations that involve multifactorial and complex processes, which are a challenge to successful cancer treatments. Accompanied by advances in biotechnology and high dimensional data analysis techniques that are bringing in new opportunities in modelling biological systems with continuous phenotypic structured models, we study a cancer cell population model that considers a multi-dimensional continuous resistance trait to multiple drugs to investigate multidrug resistance. We compare our continuous resistance trait model with classical models that assume a discrete resistance state and classify the cases when the continuum and discrete models yield different dynamical patterns in the emerging heterogeneity in response to drugs. We also compute the maximal fitness resistance trait for various continuum models and study the effect of epimutations. Finally, we demonstrate how our approach can be used to study tumour growth regarding the turnover rate and the proliferating fraction, and show that a continuous resistance level may result in a different dynamics when compared with the predictions of other discrete models.