We shall write if f(r)(t) is continuous (that is, ). If then . (It is not necessary that each of the functions f1 and f2 belongs to ). We don't distinguish functions whose difference is a smooth function. Each continuous function is equivalent (of order r=0) to 0 and each polynomial Pn(t) is equivalent to 0 with the order .
Is it true that The answer is: yes. Is it true that The answer is: no, however .(It easily may be derived that has a continuous derivative at the point t=0).