This paper presents a method to find the set of the coefficients of the transfer functions that ensure nonovershooting and monotone nondecreasing step responses. It is known that such a set is convex. In this paper, we further show that it is a closed convex set. Then, we use the convexity property of the step responses of a linear SISO system to develop a graphical method for the closed convex regions of nonovershooting and monotone nondecreasing step responses. This graphical approach is useful to aid the design of a nonovershooting or monotone nondecreasing step response controller. A servo system is used as an application example. It is found that a conventional PID feedback controller hardly ensures a nonovershooting step response. A modified PID controller is then proposed. A simulation example demonstrates the design procedure and verifies the theory.