attracting if there is a d > 0 so that for any solution x(t),
x(to) xo <8 for some to ER⇒ lim x(t) = xo.

t→∞0
Consider the onedimensional ordinary differential equation x = f(x), where f: R → R
is continuously differentiable.
(i) Show that, for any solution x(t), if f(x(to)) = 0 for some to, then f(x(t))
t€ R.
= 0 for all
(ii) Show also that, for any solution x(t), if f(x(to)) > 0 for some to, then f(x(t)) > 0 for
all t€ R; show that the same holds when > is replaced with <.
(iii) Use (i) and (ii) to show that if a critical point xo is attracting then it must also be stable.