By James R. Kirkwood

Offers advent to research of real-valued services of 1 variable. this article is for a student's first summary arithmetic direction. Writing kind is much less formal and fabric provided in a manner such that the scholar can advance an instinct for the topic and procure a few adventure in developing proofs. The slower velocity of the topic and the eye given to examples are supposed to ease the student's transition from computational to theoretical arithmetic.

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An ) = 0 for all (a1 , . . , an ) ∈ E1 × · · · × En , then show that p(x1 , . . , xn ) must be the zero polynomial, that is, all its coefficients are zero. Now suppose f is a polynomial function on a subset D of Rn , that is, suppose there is a polynomial p(x1 , . . , xn ) in n variables with coefficients in R such that f (a1 , . . , an ) = p(a1 , . . , an ) for all (a1 , . . , an ) ∈ D. Show that if D = E1 × · · · × En , where Ej is an interval containing more than one point in R for each j = 1, .

Suppose Γ lies in D, that is, (x(t), y(t)) ∈ D for all t ∈ [α, β], and Γ passes through (x0 , y0 ), that is, (x0 , y0 ) = (x(t0 ), y(t0 )) for some t0 ∈ (α, β). Given any f : D → R, the function F : [α, β] → R given by F (t) := f (x(t), y(t)) is sometimes referred to as the restriction of f to the path Γ . We shall say that f has a local maximum at (x0 , y0 ) along Γ if F has a local maximum at t0 . Likewise, we say that f has a local minimum at (x0 , y0 ) along Γ if F has a local minimum at t0 .

5. Show that if I and J are intervals in R, then the complement of their product, that is, the set D := R2 \ (I × J) = {(x, y) ∈ R2 : (x, y) ∈ I × J}, is path-connected. Further, show that if I and J are nonempty bounded intervals in R, then D is not convex. 6. Let r, s ∈ R with 0 < r < s, and let c ∈ Rn . Show that the sets Bs (c) \ Br (c) and Ss (c) \ Sr (c) are path-connected, but not convex. 7. Let k ∈ N and P1 , . . , Pk ∈ Rn . A convex combination of P1 , . . , Pk is an element in Rn of the form λ1 P1 + · · · + λk Pk , where λ1 , .