We present a mathematical model of breast cancer as a system of differential equations with piecewise constant arguments to analyze the tumor growth and chemotherapeutic treatment. We initiate a model by assuming the malignant-tumor growth under the chemotherapeutic treatment in considering the immune response by investigating the competition among both normal and tumor cells. The local stability of the system was considered by using the stability theorems for difference equations. For global stability, we assume a suitable Lyapunov function. Some sensitive parameters are considered in the stability and oscillation behaviors. To analyze the breast cancer population for the extinction case, we incorporate the Allee effect at time t. We support the theoretical results through numerical simulations.