This paper proposes new fundamental limitations on the step response of unity-feedback system based on its transfer function. Using the definition of Laplace transform, it is shown that time-domain integral equalities have to be satisfied by the step response of the SISO(single-input, single-output) stable system, and that the maximum magnitude of its step response has some lower bounds according to the achievable settling time of the system. It is also shown that the closed-loop system with open-loop RHP(right half plane) poles or zeros has some limitations in the unity-feedback control scheme. These results can be used to check in advance whether desired design specifications are achievable or not.