This paper investigates functional interval estimation for continuous-time linear systems subject to both time-varying and time-invariant uncertainties. Two novel methods are proposed based on peak-to-peak functional observer design and interval analysis. First, we present a splitting-based method that splits the estimation error dynamics into two subsystems to handle the time-invariant disturbances and provide accurate estimation results. Then, to further enhance the estimation accuracy, we present an augmentation-based method that considers the time invariance in both functional observer design and reliable interval estimation. The relationship between a state-of-art method and the proposed methods are analysed theoretically. Finally, simulation results are provided to demonstrate the performances of the proposed methods.