Question

1. Prove the following about the unit step function.

   1. (a) Multiplication by uc(t) is linear on the space of functions. That is, uc(t)(αf(t) + g(t)) = αuc(t)f(t) + uc(t)g(t).

   2. (b) uc1(t)uc2(t) = umax{c1,c2}(t).

   3. (c) uc(−t) = 1 − u−c(t) for t ̸= −c. Using a right hand limit, how can this equality be modified to hold for all t ∈ R?

   4. (d) If f : R → R is strictly increasing, then uf(c)(f(t)) = uc(t).

Answer 1