A COMMON GENERALIZATION OF THE INTERMEDIATE VALUE THEOREM AND ROUCHÉ’S THEOREM

Richard Bayne, Terrence Edwards, and Myung H. Kwack

Abstract. A simple proof of a theorem unifying Bolzano’s Theorem [8], the Intermediate Value Theorem, Rouché’s Theorem [3] and its extensions to differentiable maps to ℝ^{n} [2, 6, 9] is obtained. This unifying theorem in particular shows that in Professor Baker’s [1] examples where the number of solutions of f(x) = y for a continuous map f: B^{2} → ℝ^{2}, y ∉ f(∂B^{2}), from the unit ball B^{2} in the plane ℝ^{2} is not exactly the absolute value of the winding number of the curve f(∂B^{2}) about y, the number of the connected components of the solution set counted with multiplicity coincides with the winding number.

1. Introduction. Professor Shih [8] has observed that Bolzano’s Theorem, an equivalent of the Intermediate Value Theorem, may be stated as follows: If f is a real-valued continuous function on the closed interval I = [−1, 1] and xf(x) > 0 for x ∈ ∂I, the boundary of I, then f has at least one zero in I. Theorem A presents an analogue of Bolzano’s Theorem [8] and Rouché’s Theorem [1, 3] for the n-dimensional complex plane ℂ^{n}.

\underline{Theorem A}. Let Ω be a bounded domain in ℂ^{n} and let f, g: \overline{Ω} → ℂ^{n} be continuous and holomorphic in Ω. Then

1. [9] The map f has exactly one zero in Ω if the origin O ∈ Ω and Re(\overline{z} · f(z)) > 0 for z ∈ ∂Ω.

2. [2, 6] The maps f and g have the same number of zeros counting

multiplicity if

|f(z) − g(z)| < |f(z)| for z ∈ ∂Ω.

In this note, using the notion of intersection number defined for continuous maps [7], a simple proof of a unifying generalization of Theorem A to continuous maps in higher dimensions is obtained. The proof is simple, direct, and accessible.

2. Preliminaries. Definitions and terminologies are adopted from Guillemin and Pollack [4]. Let f: X → Y be a smooth map from a manifold X with boundary ∂X to a manifold Y. Submanifolds and the boundary of a manifold with orientation are as usual provided with induced orientations. A point y ∈ Y is a regular value of f: X → Y if df_{x}(T_{x}(X)) = T_{y}(Y) for x ∈ f^{−1}(y), where df_{x} denotes the differential map of T_{x}(X), the tangent space of X at x. The set of regular values of f, R(f), is dense in Y by Sard’s Theorem [4]. Suppose X ⊂ ℝ^{n} is a domain and A ⊂ \overline{X}. The maps f, g: \overline{X} → Y are (smoothly in X) [mod (A, y)] homotopic if there is a continuous map F: (α, β) × \overline{X} → Y, [0, 1] ⊂ (α, β), (smooth in (α, β) × X), [y ∈ Y −F ([0, 1]×A)] with F (0, z) = f(z) and F (1, z) = g(z). If f: X → Y is a map and D ⊂ X, the restriction of f to D will be denoted by f_{D}.

\underline{Definition 1}. Let Ω ⊂ ℝ^{n} be a bounded domain and let f: \overline{Ω} → ℝ^{n} be continuous and differentiable in Ω with y ∈ ℝ^{n} − f(∂Ω). The intersection number, I(f, y), of f(Ω) with y is defined as lim sup_{y_{i}→y}{N(f, y_{i}) : y_{i} ∈ R(f)}, where N(f, y_{i}) denotes the total number of preimages x ∈ f^{−1}(y_{i}) counting orientation, i.e., with a preimage x making a contribution +1 or −1 depending whether the determinant of (df)_{x} is positive or negative, respectively.

\underline{Proposition 1}. Let Ω ⊂ ℝ^{n} be a bounded domain and let f: \overline{Ω} → ℝ^{n} be continuous and differentiable in Ω with y ∈ ℝ^{n} −f(∂Ω). Then I(f, w) = I(f, y) for w ∈ W, the component of ℝ^{n} − f(∂Ω) containing y.

\underline{Proof}. Since W ∩R(f) is connected and dense in W, it suffices to show that the map h defined by h(z) = N(f, z) is locally constant in W ∩ R(f). Let z ∈ W ∩ R(f). If f^{−1}(z) = ∅, then there is a neighborhood V ⊂ W of z such that f^{−1}(V) = ∅ and thus, h(w) = 0 for w ∈ V. Suppose f^{−1}(z) = {x_{1}, …, x_{k}}. There is an open neighborhood V ⊂ W of z and open neighborhoods U_{i} of x_{i} such that (i) f^{−1}(V) = U_{1}∪⋯∪U_{k}, (ii) the sets \overline{U}_{i} ⊂ Ω are pairwise disjoint, and (iii) each f_{i}: U_{i} → V is a diffeomorphism. This shows h(w) = h(z) for w ∈ V.

\underline{Proposition 2}. Let Ω be a bounded domain in ℝ^{n} and [0, 1] ⊂ (α, β). Let G: (α, β) × \overline{Ω} → ℝ^{n} be continuous and differentiable in (α, β) × Ω with y ∈ ℝ^{n} − G((α, β) × ∂Ω). Then I(F_{0}, y) = I(F_{1}, y), where F_{t}(x) = G(t, x).

\underline{Proof}. Let t_{0} ∈ [0, 1]. From the compactness of [0, 1], it is enough to show that there is an ϵ > 0 such that I(F_{t}, y) = I(F_{t_{0}}, y) if t ∈ (t_{0}−ϵ, t_{0}+ϵ). The map H on (α, β) × \overline{Ω} defined by H(t, x) = (t, F_{t}(x)) is differentiable and JH(t, x) = det \frac{∂H(t,z)}{∂(t,z)} = det \frac{∂F_{t}(x)}{∂(x)} for x ∈ Ω.

If H^{−1}(t_{0}, y) = ∅, there is a neighborhood V of (t_{0}, y) with (i) H^{−1}(V) = ∅ and (ii) [t_{0} − ϵ, t_{0} + ϵ] × B_{r}(y) ⊂ V for ϵ > 0 and r > 0 where B_{r}(y) = {w ∈ ℝ^{n} : |w − y| < r}. Since for t ∈ [t_{0} − ϵ, t_{0} + ϵ], F_{t}^{−1}(B_{r}(y)) = ∅, we get I(F_{t}, y) = I(F_{t_{0}}, y) = 0.

Suppose H^{−1}(t_{0}, y) ≠ ∅. Let V be a neighborhood of (t_{0}, y) with (i) H^{−1}(V) ⊂ (α, β) × Ω and (ii) [t_{0} − ϵ_{1}, t_{0} + ϵ_{1}] × B_{r′}(y) ⊂ V for ϵ_{1} > 0 and r′ > 0. Let p ∈ B_{r′}(y) ∩ R(F_{t_{0}}). If H^{−1}(t_{0}, p) = ∅, we have I(F_{t}, p) = I(F_{t_{0}}, p) = 0, as in the previous case, for t ∈ [t_{0} − ϵ, t_{0} + ϵ] with 0 < ϵ < ϵ_{1} and hence, I(F_{t}, y) = I(F_{t_{0}}, y) = 0 from Proposition 1. Now suppose H^{−1}(t_{0}, p) = {z_{1}, …, z_{k}}. Since (t_{0}, p) ∈ R(H) and so JH(t_{0}, z_{i}) ≠ 0, there are 0 < ϵ′ < ϵ_{1} and open neighborhoods U_{i} of z_{i} such that (i) \overline{U}_{i} are pairwise disjoint and (ii) H is diffeomorphic on (t_{0} − ϵ′, t_{0} + ϵ′) × U_{i}. Then F_{t} is also diffeomorphic on U_{i} for t ∈ (t_{0} − ϵ′, t_{0} + ϵ′). There are numbers r > 0 and ϵ > 0 such that (t_{0} − ϵ, t_{0} + ϵ) × \overline{B_{r}(p)} ⊂ (t_{0} − ϵ′, t_{0} + ϵ′) × B_{r′}(y) ⊂ ⋂_{i} H((t_{0} − ϵ′, t_{0} + ϵ′) × U_{i}). It follows that for t ∈ (t_{0} − ϵ, t_{0} + ϵ), F_{t}^{−1}(B_{r}(p)) ⊂ ⋃_{i} U_{i} and so I(F_{t}, p) = I(F_{t_{0}}, p) and thus from Proposition 1, I(F_{t}, y) = I(F_{t_{0}}, y).

\underline{Definition 2}. Let Ω be a bounded domain in ℝ^{n} and let f: \overline{Ω} → ℝ^{n} be continuous and differentiable in Ω with y ∈ ℝ^{n} − f(∂Ω). Let A ⊂ Ω be a component of f^{−1}(y). The order of multiplicity of f at A, μ_{A}(f), is defined by μ_{A}(f) = I(f\overline{U}, y), where U ⊂ Ω is a domain containing A with \overline{U} ∩ f^{−1}(y) = A.

\underline{Remark}. This definition agrees with that of order of multiplicity of holomorphic maps in ℂ^{n} at an isolated preimage point [2].

\underline{Theorem 3}. Let Ω ⊂ ℝ^{n} be a bounded domain and let f: \overline{Ω} → ℝ^{n} be continuous and differentiable in Ω with y ∈ ℝ^{n} − f(∂Ω). Then (1). f^{−1}(y) ≠ ∅ if I(f, y) ≠ 0. (2). I(f, y) = I(g, y) if g: \overline{Ω} → ℝ^{n} is mod (∂Ω, y) homotopic smoothly in Ω

to f. (3). I(f, y) = ∑_{i} μ_{A_{i}}(f), where {A_{1}, …, A_{k}} are the components of

f^{−1}(y).

\underline{Proof}.

(1). Assume I(f, y) ≠ 0. A sequence y_{k} ∈ R(f) converges to y and a

sequence x_{k} ∈ f^{−1}(y_{k}) has a limit point x ∈ Ω with f(x) = y. (2). This follows from Proposition 2. (3). Let U_{i} be open domains such that (i) A_{i} ⊂ U_{i} ⊂ \overline{U}_{i} ⊂ Ω, (ii) \overline{U}_{i}

are pairwise disjoint, and (iii) y ∉ f(⋃_{i} ∂U_{i} ∪ (∂Ω)). Let y_{k} ∈ R(f) be a sequence converging to y such that f^{−1}(y_{k}) ⊂ ⋃ U_{i}. Now I(f, y) = lim sup_{k} I(f, y_{k}) = lim sup_{k} ∑_{i} I(f_{\overline{U}_{i}}, y_{k}) = ∑_{i} I(f_{\overline{U}_{i}}, y) = ∑_{i} μ_{A_{i}}(f).

3. Main Results. An extension of the definition of intersection number to continuous f: \overline{Ω} → ℝ^{n} with y ∈ ℝ^{n} − f(∂Ω), where Ω is a bounded domain in ℝ^{n} is now given. By the Stone-Weierstrass Theorem [5] there is a sequence of polynomials P_{j}: \overline{Ω} → ℝ^{n} which converges uniformly to f. The following lemma will facilitate a proof of Theorem 4, extending the collection of maps in Theorem 3 to a collection of continuous maps.

\underline{Convergence Lemma}. Let Ω be a bounded domain in ℝ^{n} and let f: \overline{Ω} → ℝ^{n} be continuous with y ∈ ℝ^{n} − f(∂Ω). Suppose P_{k}, Q_{j} are sequences of polynomials converging uniformly to f on \overline{Ω}. Then I(P_{j}, y) = I(Q_{k}, y) for j, k ≥ N for a number N.

\underline{Proof}. Let r = inf_{z∈∂Ω}\{|y − z| > 0\}. There is an integer N such that

sup\{|g(x) − f(x)| : g(x) = P_{k}(x) or g(x) = Q_{j}(x), x ∈ ∂Ω, j, k > N\} < \frac{r}{4}.

Let k, j > N and define F_{k,j}: [0, 1] × \overline{Ω} → ℝ^{n} by F_{k,j}(t, x) = tP_{k}(x) + (1 − t)Q_{j}(x). Since y ∉ F_{k,j}([0, 1] × ∂Ω), F_{k,j} defines a smoothly in Ω mod (∂Ω, y) homotopy of the polynomials P_{k}, Q_{j}: \overline{Ω} → ℝ^{n}. The conclusion follows from Theorem 3(2).

By the Convergence Lemma the notions of intersection number and order of multiplicity may be extended to continuous maps.

\underline{Definition 3}. Let Ω be a bounded domain in ℝ^{n} and let f: \overline{Ω} → ℝ^{n} be a continuous map. Let y ∈ ℝ^{n} − f(∂Ω) and let A be a component of f^{−1}(y). (1). The intersection number, I(f, y), of f(Ω) with y is defined by I(f, y) =

lim sup I(P_{i}, y), where P_{i}: \overline{Ω} → ℝ^{n} is a sequence of polynomials which converges uniformly to f.

(2). The order of multiplicity, μ_{A}(f), of f at A is defined by μ_{A}(f) = lim sup I(f\overline{U}, y), where U ⊂ Ω is a domain containing A with \overline{U} ∩ f^{−1}(y) = A.

Theorem 4 below, the main result, is a common generalization of Bolzano’s Theorem, the Intermediate Value Theorem, Rouché’s Theorem and Theorem A as Corollary 5 demonstrates.

\underline{Theorem 4}. Let Ω ⊂ ℝ^{n} be a bounded domain and let f: \overline{Ω} → ℝ^{n} be continuous with y ∈ ℝ^{n} − f(∂Ω). Then (1). f^{−1}(y) ≠ ∅ if I(f, y) ≠ 0. (2). I(f, y) = I(g, y) if g: \overline{Ω} → ℝ^{n} is continuous and ∂f, ∂g: ∂Ω → ℝ^{n} − \{y\}

are homotopic. (3). I(f, y) = ∑_{i} μ_{A_{i}}(f), where {A_{1}, …, A_{k}} are the components of

f^{−1}(y).

\underline{Proof}.

(1). Suppose P_{j}: \overline{Ω} → ℝ^{n} is a sequence of polynomials which converges uniformly to f such that I(P_{j}, y) = I(f, y). Then I(P_{j}, y) ≠ 0 for all j and so for a sequence x_{j} ∈ Ω, P_{j}(x_{j}) = y. Then f(x) = y where x ∈ Ω is a limit point of a convergent subsequence of x_{j}.

(2). Let G: ([0, 1]×∂Ω) → ℝ^{n}−{y} be a homotopy of ∂f, ∂g: ∂Ω → ℝ^{n}−{y} with G(0, x) = ∂f(x) and G(1, x) = ∂g(x). Define a map F on ({0, 1}× \overline{Ω}) ∪ ([0, 1] × ∂Ω) by F (t, x) = G(t, x) for (t, x) ∈ ([0, 1] × ∂Ω) and for x ∈ \overline{Ω}, F (0, x) = f(x) and F (1, x) = g(x). By the Stone-Weierstrass Theorem there is a sequence of polynomials F_{k}: \overline{Ω} → ℝ^{n} converging uniformly to F on (\{0, 1\}×\overline{Ω})∪([0, 1]×∂Ω). Define Q_{k,t,t′}: [0, 1]×\overline{Ω} → ℝ^{n} for t, t′ ∈ [0, 1] by Q_{k,t,t′}(s, x) = sF_{k}(t, x)+(1−s)F_{k}(t′, x). For each t_{0} ∈ [0, 1] there is an ϵ > 0 such that for t, t′ ∈ [t_{0} − ϵ, t_{0} + ϵ], we have y ∉ Q_{k,t,t′}([0, 1]×∂Ω) eventually, say for k ≥ L. Let t, t′ ∈ [t_{0}−ϵ, t_{0}+ϵ]. Then Q_{k,t,t′} defines a smoothly in Ω mod (∂Ω, y) homotopy of the polynomials P_{k,t}, P_{k,t′}: \overline{Ω} → ℝ^{n}, where P_{k,t}(x) = F_{k}(t, x). By the compactness of [0, 1] the maps P_{k,0}, P_{k,1} are mod (∂Ω, y) homotopic. The conclusion follows from Theorem 3(2).

(3). Let W_{i} ⊂ Ω be domains containing A_{i} with \overline{W}_{i} ∩ f^{−1}(y) = A_{i} and with pairwise disjoint closures. Suppose P_{j}: \overline{Ω} → ℝ^{n} is a sequence of polynomials converging uniformly to f such that for an integer L and j ≥ L, I(P_{j}, y) = I(f, y) and I((P_{j})_{\overline{W}_{i}}, y) = I(f_{\overline{W}_{i}}, y). By Theorem 3, I(P_{j}, y) = ∑_{i} I((P_{j})_{\overline{W}_{i}}, y) and thus I(f, y) = ∑_{i} I(f_{\overline{W}_{i}}, y).

Corollary 5 extends Theorem A to continuous maps.

\underline{Corollary 5}. Let Ω be a bounded domain in ℝ^{n} and let f, g: \overline{Ω} → ℝ^{n} be continuous. Then f^{−1}(O) and g^{−1}(O) have the same number of components when counted with multiplicity if any one of the following is satisfied for z ∈ ∂Ω.

i. |f(z) + g(z)| < |f(z)| + |g(z)|. ii. |f(z) − g(z)| < |f(z)|. iii. Re(\overline{f(z)} · g(z)) > 0, where ℝ^{n} ≠ ℂ^{k}.

\underline{Proof}. If any one of the conditions is satisfied for z ∈ ∂Ω, the vectors g(z) and f(z) are not collinear and so the map F (t, z) = tf(z) + (1 − t)g(z) defines a mod (∂Ω, O) homotopy of f and g. Theorem 4 finishes the proof.

Corollary 6 extends Theorem A(1).

\underline{Corollary 6}. Let Ω be a bounded domain in ℂ^{n} containing O and let f: \overline{Ω} → ℂ^{n} be a continuous map. If Re(\overline{z} · f(z)) > 0 for z ∈ ∂Ω, then I(f, O) = 1 and so f^{−1}(O) ≠ ∅. In addition, if f is holomorphic in Ω, then f^{−1}(O) is a singleton.

\underline{Proof}. The first part is immediate from Corollary 5 and Theorem 4(1). The second part follows from the first and the fact that a compact analytic subset of ℂ^{n} is a finite set and an order of multiplicity for a holomorphic map is nonnegative.

Professor Baker [1] presented examples of maps below where the number of solutions of f(x) = y for a continuous map f: B^{2} → ℝ^{2}, y ∉ f(∂B), is not exactly the absolute value, |γ(f(∂B^{2}), y)|, of the winding number of the curve f(∂B^{2}) about y. Theorem 4 shows that if each connected component of the solution set is counted with multiplicity, the total number coincides with the winding number.

\underline{Examples}. Let f, g, h be maps on the unit disk B^{2} defined by f(x, y) = (x, |y|), g(x, y) = (x^{2} − y^{2}, −2xy) and h(x, y) = ϕ(x^{2} + y^{2})(x, y), where

ϕ(t) = \{{0, if 0 ≤ t ≤ \frac{1}{2}; \atop 2t − 1, if \frac{1}{2} ≤ t ≤ 1 .}

(1). The winding number γ(f(∂B^{2}), (0, 0)) = 0. The equation f(x, y) = (0, 0) has exactly one solution (0, 0) with μ_{(0,0)}(f) = 0 and the equation f(x, y) = (0, \frac{1}{2}) has exactly two solutions with μ_{(0, \frac{1}{2} )}(f) = 1 and μ_{(0,− \frac{1}{2} )}(f) = −1.

(2). The winding number γ(g(∂B^{2}), (0, 0)) = −2. The equation g(x, y) = (0, 0) has exactly one solution (0, 0) with μ_{(0,0)}(g) = −2 and the equation g(x, y) = (\frac{1}{4}, 0) has exactly two solutions (0, 0) with μ_{( \frac{1}{2} ,0)}(g) = −1 and μ_{(− \frac{1}{2} ,0)}(g) = −1.

(3). The winding number γ(h(∂B^{2}), (0, 0)) = 1. The solution of the equation h(x, y) = (0, 0) is the connected set A = \{(x, y) : |(x, y)| ≤ \frac{1}{2}\} with μ_{A}(h) = 1.

\underline{References}

1. J. A. Baker, “Plane Curves, Polar Coordinates and Winding Numbers,” Mathematics Magazine, 64 (1991), 75–91.

2. E. M. Chirka, Complex Analytic Sets, Mathematics and Its Applications, Kluwer Academic Publishers, Boston, 1989.

3. I. Glicksberg, “A Remark on Rouché’s Theorem,” Amer. Math. Monthly, 83 (1976), 186.

4. V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., Englewood, NJ, 1974.

5. S. Lang, Real Analysis, Addison-Wesley Publishing Co., Reading, MA, 1983.

6. N. G. Lloyd, “Remarks on Generalizing Rouché’s Theorem,” J. London Math. Soc., 20 (1979), 259–272.

7. M. Nagumo, “Degree of Mapping of Manifolds Based on That of Euclidean Open Sets,” Osaka Math. J., 2 (1950), 105–118.

8. M. H. Shih, “An Analog of Bolzano’s Theorem for Functions of a Complex Variable,” Amer. Math. Monthly, 89 (1982), 210–211.

9. M. H. Shih, “Bolzano’s Theorem in Several Complex Variables,” Proc. Amer. Math. Soc., 79 (1982), 32–34.

Mathematics Subject Classification (2000): 30C15, 55M25

Richard Bayne Department of Mathematics Howard University Washington, D. C. 20059 email: rbayne@howard.edu

Terrence Edwards Department of Mathematics Dillard University New Orleans, LA 70122 email: tajedwards@netscape.net

Myung H. Kwack Department of Mathematics Howard University Washington, D. C. 20059 email: mkwack@howard.edu