4.

(a) Let c E R, 0> 0, and set T(U) = ou + c for U E R. Prove that if (¢, I) is a smooth parametrization of some curve, if J = T- 1 (1), and if 'if; = ¢ 0 T, then

('if;, J) is orientation equivalent to (¢,1).

(b) Prove that if (¢, 1) is a parametrization of some smooth arc, then it has an orientation equivalent parametrization of the form ('if;, [0, 1]).

(c) Obtain an analogue of

(b) for piecewise smooth curves.