Pulse Function For positive numbers a < b , the pulse function is defined as

P a , b ( x ) = H ( x − a ) − H ( x − b ) = { 0 , x < a 1 , a ≤ x < b 0 , x ≥ b

where H ( x ) = { 1 , x ≥ 0 0 , x < 0 is the Heaviside function.

(a) Sketch the graph of the pulse function.

(b) Find the following limits:

(i) lim x → a + P a , b ( x )

(ii) lim x → a + P a , b ( x )

(iii) lim x → b + P a , b ( x )

(iv) lim x → b − P a , b ( x )

(c) Discuss the continuity of the pulse function.

(d) Why is U ( x ) = 1 b − a P a , b ( x ) called the unit puke function?