Generalized Functions: Theory and Technique (Enhanced Edition‪)‬

The Heaviside function H(x) is defined to be equal to zero for every negative value of x and to unity for every positive value of x; that is, H(x) = 0, x is less than 0, 1, x > 0. It has a jump discontinuity at x = 0 and is also called the unit step function. Its value at x = 0 is usually taken to be ½. Sometimes it is taken to be a constant c, 0 is less than c is less than 1, and then the function is written Hc(x). If the jump in the Heaviside function is at a point x = a, then it is written H(x-a). Observe that H(x-1) = 1-H(x), H(a-x) = 1-H(x-a). (2) The functions H(x) , H(x-a), and H(a-x) are drawn in Fig. 1.1.