In this paper, the controllable frequency-tuning and laser output characteristics of a single-longitudinal-mode distributed Bragg reflector (DBR) fiber laser with a piezoelectric ceramic that axially stretches across a DBR fiber resonant cavity are investigated. Our theoretical analysis shows that this tuning scheme can effectively prevent the occurrence of mode hopping during frequency tuning, which is beneficial for obtaining a wider frequency-tuning range without mode hopping. Additionally, experimental results verify that this tuning mechanism can achieve fast and controllable frequency tuning of the DBR fiber laser over a wide range without mode hopping. The maximum mode-hop-free frequency-tuning range obtained in our experiment is as high as 116 GHz. During the frequency-tuning process, changes in the linewidth and relative intensity noise of the laser are not significant; furthermore, large fluctuations in the output laser power can be effectively suppressed using a boost fiber-optic amplifier.