(a) Use the step-down function $$u_c(t)$$ to construct a step-up and a step-up, step-down function.

(b) Prove that $${\cal L}\{u_c(t) f(t-c)\} = e^{-cs}F(s)$$.

(c) Consider the piecewise continuous function given by

$$f(t) =\begin{cases}t, & \text{if } t<1;\\1, & \text{if } t\ge 1.\end{cases}$$

(i) Express $$f(t)$$ in a single line using the Heaviside function.

(ii) Find $$F=F(s)$$.