Let ∫ = a_nxⁿ + · · · + a₀ be a polynomial over the field R of real numbers and Jet φ = |a_n|xⁿ + · · · + |a_o| ϵ R[x].

(a) If |u| ≤ d, then |f(u)| ≤ φ(d). [Recall that |a + b| ≤ |a| + |b| and that |a| ≤ a', |b|≤ b ⇒ |ab| ≤ a'b'.]

(b) Given a,c ϵ R with c > 0 there exists M ϵ R such that |∫(a + h) - ∫(a)| ≤ M|h| for all h ϵ R with |h|≤ c.

(c) (Intermediate Value Theorem) If a < b and ∫(a) < d < f(b), then there exists c ϵ R such that a < c < b and ∫(c) = d.

(d) Every polynomial g of odd degree in R[x] has a real root.