On the Single-Valuedness and Multi-Valuedness of Functions of a Real Variable

Tiberiu Popoviciu
(proceedings), Mathematical Analysis, paper

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T. Popoviciu
Institutul de Calcul

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T. Popoviciu, Über die Einwertigkeit und die Mehrwertigkeit der Funktionen einer reellen Variablen, IKM, IV Internationaler Kongress über Anwendungen der Mathematik in dem ingenieurwissenschaften, Berichte 2, Weimar (DDR) 1967, pp. 155-158 (in German).

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1967 a -Popoviciu- IKM Berlin - On the univalence and multivalence of the functions of an r
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IKM, IV International Congress on Applications of Mathematics in Engineering, Reports 2, Weimar 1967.

Popoviciu, T. 1 ) 1 ) ^(1)){ }^{1)}1)

ON THE UNIVALENCE AND MULTIVALENCE OF FUNCTIONS OF A REAL VARIABLE

  1. In the approximation theory of functions and its applications, special attention is paid to certain special classes of functions for which the corresponding approximation problems exhibit certain special aspects. For example, the approximation by polynomials of continuous, differentiable, or integrable functions raises specific problems for each of these function classes, which are well known in analysis. In my presentation, I will focus particularly on those properties that characterize certain behaviors of functions. It is difficult to give a precise definition of the concept of behavior or a specific behavior of a function; nevertheless, such properties are encountered at every turn. The behavior of a function is understood, for example, as the way in which it changes, or more intuitively, as its graphical representation. In what follows, we will limit ourselves to a few examples of the above.
  2. Let a function be defined in the interval I. One property of this function is its non-negativity (in particular, its positivity). These functions have the property that their set forms a wedge (or a non-negative linear set). In other words, every linear combination with non-negative coefficients of a finite number of non-negative functions is itself a non-negative function. A property related to non-negativity is bounded below. The functions bounded below in I also form a wedge. For a function f f fffis bounded below, it is necessary and sufficient that there exists a constant C such that the function f ( x ) − C f ( x ) − C f(x)-Cf(x)-Cf(x)−Cis non-negative.
  3. Another important wedge is formed by the non-decreasing functions in I. A similar property is enjoyed by the functions f f fff, for which there is a first-degree polynomial ax + b + b +b+b+bso that the function f ( x ) − a x − b f ( x ) − a x − b f(x)-ax-bf(x)-a x-bf(x)−ax−bis non-decreasing. These are the functions for which the slope
[ x 1 , x 2 ; f ] = f ( x 2 ) − f ( x 1 ) x 2 − x 1 x 1 , x 2 ; f = f x 2 − f x 1 x 2 − x 1 [x_(1),x_(2);f]=(f(x_(2))-f(x_(1)))/(x_(2)-x_(1))\left[x_{1}, x_{2} ; f\right]=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}[x1,x2;f]=f(x2)−f(x1)x2−x1
is bounded below. Functions that have a slope bounded below also form a wedge.
4. A special case of non-decreasing functions are (strictly) increasing functions. A continuously increasing or decreasing, i.e., strictly monotonic function in the interval I, can also be characterized by its single-valuedness, by the property that each of its values ​​is assumed only once. Consequently, we have the following
Theorem 1. For a continuous function defined in the interval I to be strictly monotonic (increasing or decreasing), it is necessary and sufficient that it be single-valued.
The condition is necessary. If the function f f fffwould not be univalent, there would be two different points x 1 , x 2 ∈ I x 1 , x 2 ∈ I x_(1),x_(2)in Ix_{1}, x_{2} \in Ix1,x2∈Iso that f ( x 1 ) = f ( x 2 ) f x 1 = f x 2 f(x_(1))=f(x_(2))f\left(x_{1}\right)=f\left(x_{2}\right)f(x1)=f(x2), and then we would have [ x 1 , x 2 ; f ] = 0 x 1 , x 2 ; f = 0 [x_(1),x_(2);f]=0\left[x_{1}, x_{2} ; f\right]=0[x1,x2;f]=0and the function might not be strictly monotonic.
The condition is sufficient. Let us assume that the function is continuous and single-valued. We will show that f f f\mathbf{f}fis strictly monotonic. If this were not the case, we would have to examine the following two cases.
  1. There are two different points x 1 , x 2 ∈ I x 1 , x 2 ∈ I x_(1),x_(2)in Ix_{1}, x_{2} \in Ix1,x2∈I, so that [ x 1 , x 2 ; f ] = 0 x 1 , x 2 ; f = 0 [x_(1),x_(2);f]=0\left[x_{1}, x_{2} ; f\right]=0[x1,x2;f]=0. It follows f ( x 1 ) = f ( x 2 ) f x 1 = f x 2 f(x_(1))=f(x_(2))f\left(x_{1}\right)=f\left(x_{2}\right)f(x1)=f(x2), which contradicts the univalence of the function.
  2. There are different points x 1 ′ , x 2 ′ ∈ I x 1 ′ , x 2 ′ ∈ I x_(1)^('),x_(2)^(')in Ix_{1}^{\prime}, x_{2}^{\prime} \in Ix1′,x2′∈Iand the different points x 1 ′ ′ , x 2 ′ ′ ∈ I x 1 ′ ′ , x 2 ′ ′ ∈ I x_(1)^(''),x_(2)^('')in Ix_{1}^{\prime \prime}, x_{2}^{\prime \prime} \in Ix1′′,x2′′∈I, so that
[ x 1 ′ , x 2 ′ ; f ] = A < 0 [ x 1 ′ ′ , x 2 ′ ′ ; f ] = B > 0 x 1 ′ , x 2 ′ ; f = A < 0 x 1 ′ ′ , x 2 ′ ′ ; f = B > 0 [x_(1)^('),x_(2)^(');f]=A < 0quad[x_(1)^(''),x_(2)^('');f]=B > 0\left[\mathrm{x}_{1}^{\prime}, \mathrm{x}_{2}^{\prime} ; \mathrm{f}\right]=\mathrm{A}<0 \quad\left[\mathrm{x}_{1}^{\prime \prime}, \mathrm{x}_{2}^{\prime \prime} ; \mathrm{f}\right]=\mathrm{B}>0[x1′,x2′;f]=A<0[x1′′,x2′′;f]=B>0
We accept x 1 ′ < x 2 ′ , x 1 ′ ′ < x 2 ′ ′ x 1 ′ < x 2 ′ , x 1 ′ ′ < x 2 ′ ′ x_(1)^(') < x_(2)^('),x_(1)^('') < x_(2)^('')x_{1}^{\prime}<x_{2}^{\prime}, x_{1}^{\prime \prime}<x_{2}^{\prime \prime}x1′<x2′,x1′′<x2′′and set
x 1 = x 1 ( l ) = l x 1 ′ + ( 1 − l ) x 1 ′ ′ , x 2 = x 2 ( l ) = l x 2 ′ ′ + ( 1 − l ) x 2 ′ ′ . x 1 = x 1 ( l ) = l x 1 ′ + ( 1 − l ) x 1 ′ ′ , x 2 = x 2 ( l ) = l x 2 ′ ′ + ( 1 − l ) x 2 ′ ′ . x_(1)=x_(1)(lambda)=lambdax_(1)^(')+(1-lambda)x_(1)^(''),x_(2)=x_(2)(lambda)=lambdax_(2)^('')+(1-lambda)x_(2)^('').x_{1}=x_{1}(\lambda)=\lambda x_{1}^{\prime}+(1-\lambda) x_{1}^{\prime \prime}, x_{2}=x_{2}(\lambda)=\lambda x_{2}^{\prime \prime}+(1-\lambda) x_{2}^{\prime \prime} .x1=x1(l)=lx1′+(1−l)x1′′,x2=x2(l)=lx2′′+(1−l)x2′′.
Then [ x 1 , x 2 ; f ] x 1 , x 2 ; f [x_(1),x_(2);f]\left[x_{1}, x_{2} ; f\right][x1,x2;f]one in the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]continuous function of l l lambda\lambdal, which it it it\operatorname{den}itValue A A AAA ful l = 1 l = 1 lambda=1\lambda=1l=1and the
Value B for l = 0 l = 0 lambda=0\lambda=0l=0It follows that there is a value l ϵ ( 0 , 1 ) l ϵ ( 0 , 1 ) lambda epsilon(0,1)\lambda \epsilon(0,1)lϵ(0,1)for a x 1 ≠ x 2 x 1 ≠ x 2 x_(1)!=x_(2)\mathrm{x}_{1} \neq \mathrm{x}_{2}x1≠x2and [ x 1 , x 2 ; f ] = 0 x 1 , x 2 ; f = 0 [x_(1),x_(2);f]=0\left[\mathrm{x}_{1}, \mathrm{x}_{2} ; \mathrm{f}\right]=0[x1,x2;f]=0, and thus we return to the first case.
This proves Theorem 1.
5. The question now arises: how could one characterize the behavior of a function if it can take some of its values ​​several times? For simplicity, we will say that a function in We have the following
Theorem 2. If a continuous function defined in a finite and closed interval I is two-valued, then it is possible to partition the interval I into at most three closed consecutive intervals such that in each of them the function is (strictly) monotonic.
One says that the successive intervals I 1 , I 2 , … , I k I 1 , I 2 , … , I k I_(1),I_(2),dots,I_(k)I_{1}, I_{2}, \ldots, I_{k}I1,I2,…,Ik, if for each a = 1 , 2 , … , k − 1 a = 1 , 2 , … , k − 1 alpha=1,2,dots,k-1\alpha=1,2, \ldots, k-1a=1,2,…,k−1the Intervals I α , I α + 1 I α , I α + 1 I_(alpha),I_(alpha+1)I_{\alpha}, I_{\alpha+1}Ia,Ia+1a single common point.
, It is I = [ a , b ] I = [ a , b ] I=[a,b]I=[a, b]I=[a,b], ( a < b a < b a < ba<ba<b), and we assume the function f f fffis not a constant and it is two-valued in [ a , b ] [ a , b ] [a,b][a, b][a,b]. From the continuity of f f ffffollows the existence of a c ∈ [ a , b ] c ∈ [ a , b ] c in[a,b]c \in[a, b]c∈[a,b]and one d ϵ [ a , b ] d ϵ [ a , b ] d epsilon[a,b]d \epsilon[a, b]dϵ[a,b]so that inf f = f ( c ) f = f ( c ) f=f(c)f=f(c)f=f(c), sup f = f ( α ) f = f ( α ) f=f(alpha)f=f(\alpha)f=f(a). Without loss of generality, we can say that d [c,d] is single-valued. If this were not the case, there would be two points x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2so that c ≤ x 1 < x 2 ≤ d c ≤ x 1 < x 2 ≤ d c <= x_(1) < x_(2) <= dc \leq x_{1}<x_{2} \leq dc≤x1<x2≤dand f ( x 1 ) = f ( x 2 ) f x 1 = f x 2 f(x_(1))=f(x_(2))f\left(x_{1}\right)=f\left(x_{2}\right)f(x1)=f(x2), also [ x 1 , x 2 , f ] = 0 x 1 , x 2 , f = 0 [x_(1),x_(2),f]=0\left[x_{1}, x_{2}, f\right]=0[x1,x2,f]=0. Based on the mean value theorem Wör x 1 , x 2 ′ x 1 , x 2 ′ x_(1),x_(2)^(')x_{1}, x_{2}^{\prime}x1,x2′find so that x 1 < x 1 < x 2 ′ < x 2 x 1 < x 1 < x 2 ′ < x 2 x_(1) < x_(1) < x_(2)^(') < x_(2)x_{1}<x_{1}<x_{2}^{\prime}<x_{2}x1<x1<x2′<x2and [ x 1 ′ , x 2 ′ ] x 1 ′ , x 2 ′ [x_(1)^('),x_(2)^(')]\left[x_{1}^{\prime}, x_{2}^{\prime}\right][x1′,x2′].
can we f ( x 1 ′ ) = f ( x 2 ′ ) < f ( x 1 ) , f ( x 1 ) = f ( x 2 ) f x 1 ′ = f x 2 ′ < f x 1 , f x 1 = f x 2 f(x_(1)^('))=f(x_(2)^(')) < f(x_(1))quad,f(x_(1))=f(x_(2))f\left(x_{1}^{\prime}\right)=f\left(x_{2}^{\prime}\right)<f\left(x_{1}\right) \quad, f\left(x_{1}\right)=f\left(x_{2}\right)f(x1′)=f(x2′)<f(x1),f(x1)=f(x2). Without limiting the generality, f ( x 1 ) = f ( x 2 ) < f ( x 1 ) = f ( x 2 ) f x 1 = f x 2 < f x 1 = f x 2 f(x_(1))=f(x_(2)) < f(x_(1))=f(x_(2))\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{f}\left(\mathrm{x}_{2}\right)<\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{f}\left(\mathrm{x}_{2}\right)f(x1)=f(x2)<f(x1)=f(x2).
f ( x ) , y , x , y f ( x ) , y , x , y f(x),y,x,y\mathrm{f}(\mathrm{x}), \mathrm{y}, \mathrm{x}, \mathrm{y}f(x),and,x,andassumed. So f ( x 0 ) = f ( x 1 ) = f ( x 2 ) f x 0 = f x 1 = f x 2 f(x_(0))=f(x_(1))=f(x_(2))f\left(x_{0}\right)=f\left(x_{1}\right)=f\left(x_{2}\right)f(x0)=f(x1)=f(x2), which contradicts the bivalence of the function. Likewise, the case
f ( x 1 ′ ) = f ( x 2 ′ ) > f ( x 1 ) = f ( x 2 ) f x 1 ′ = f x 2 ′ > f x 1 = f x 2 f(x_(1)^('))=f(x_(2)^(')) > f(x_(1))=f(x_(2))f\left(x_{1}^{\prime}\right)=f\left(x_{2}^{\prime}\right)>f\left(x_{1}\right)=f\left(x_{2}\right)f(x1′)=f(x2′)>f(x1)=f(x2)
We have thus proven that the function f f fffin the interval [ c , d ] [ c , d ] [c,d][c, d][c,d]By Theorem 1, it is therefore monotone in [ c , d ] [ c , d ] [c,d][c, d][c,d]. Due to the definition of the points c , d c , d c,dc, dc,dand the property that the function is two-valued, it follows that the function in each of the intervals [ a , c ] , [ d , b ] [ a , c ] , [ d , b ] [a,c],[d,b][a, c],[d, b][a,c],[d,b]is single-valued, and consequently it is monotonic in these intervals.
Likewise, the case a ≦ d < c ≦ b a ≦ d < c ≦ b a <= d < c <= b\mathrm{a} \leqq \mathrm{d}<\mathrm{c} \leqq \mathrm{b}a≦d<c≦btreated, and thus Theorem 2 is proven.
6. A function is called three-valued if it takes each of its values ​​at most three times. There is no theorem corresponding to Theorem 2 for three-valued functions, in the sense that for a continuous and three-valued function, the interval I cannot in general be partitioned into a finite number of consecutive monotonicity intervals.
To prove this, it suffices to show a continuous function defined in the interval [ 0,1 ], for which
f ( 0 ) = 0 , f ( 1 2 n − 1 ) ; f ( 3 2 n + 2 ) = 3 2 n + 2 , n = 0 , 1 , … f ( 0 ) = 0 , f 1 2 n − 1 ; f 3 2 n + 2 = 3 2 n + 2 , n = 0 , 1 , … f(0)=0,f((1)/(2^(n-1)));f((3)/(2^(n+2)))=(3)/(2^(n+2)),n=0,1,dotsf(0)=0, f\left(\frac{1}{2^{n-1}}\right) ; f\left(\frac{3}{2^{n+2}}\right)=\frac{3}{2^{n+2}}, n=0,1, \ldotsf(0)=0,f(12n−1);f(32n+2)=32n+2,n=0,1,…
and which is linear in the intervals
[ 1 2 n + 1 , 3 2 n + 2 ] , [ 3 2 n + 2 , 1 2 n ] , n = 0 , 1 , … ist. 1 2 n + 1 , 3 2 n + 2 , 3 2 n + 2 , 1 2 n , n = 0 , 1 , …  ist.  [(1)/(2^(n+1)),(3)/(2^(n+2))],[(3)/(2^(n+2)),(1)/(2^(n))],n=0,1,dots" ist. "\left[\frac{1}{2^{\mathrm{n}+1}}, \frac{3}{2^{\mathrm{n}+2}}\right],\left[\frac{3}{2^{\mathrm{n}+2}}, \frac{1}{2^{\mathrm{n}}}\right], \mathrm{n}=0,1, \ldots \text { ist. }[12n+1,32n+2],[32n+2,12n],n=0,1,… is. 
Image 2

[ 1 2 n + 1 3 2 n + 2 ] = 0 , 1 … 1 2 n + 1 3 2 n + 2 = 0 , 1 … [(1)/(2^(n+1))(3)/(2^(n+2))]=0,1dots\left[\frac{1}{2^{n+1}} \frac{3}{2^{n+2}}\right]=0,1 \ldots[12n+132n+2]=0,1…

  1. The above results can be generalized in several directions by seeking a corresponding generalization of the monotonicity property of a function. Such a generalization is obtained by introducing the concept of a higher-order convex function. To do this, one first introduces the higher-order gradients using the recursion formula
    [ x 1 , x 2 , … , x n + 1 ; f ] = [ x 2 , x 3 , … , x n + 1 ; f ] − [ x 1 , x 2 , … , x n ; f ] x n + 1 − x 1 x 1 , x 2 , … , x n + 1 ; f = x 2 , x 3 , … , x n + 1 ; f − x 1 , x 2 , … , x n ; f x n + 1 − x 1 [x_(1),x_(2),dots,x_(n+1);f]=([x_(2),x_(3),dots,x_(n+1);f]-[x_(1),x_(2),dots,x_(n);f])/(x_(n+1)-x_(1))\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\frac{\left[x_{2}, x_{3}, \ldots, x_{n+1} ; f\right]-\left[x_{1}, x_{2}, \ldots, x_{n} ; f\right]}{x_{n+1}-x_{1}}[x1,x2,…,xn+1;f]=[x2,x3,…,xn+1;f]−[x1,x2,…,xn;f]xn+1−x1
[ x 1 ; f ] = f ( x 1 ) x 1 ; f = f x 1 [x_(1);f]=f(x_(1))\left[x_{1} ; f\right]=f\left(x_{1}\right)[x1;f]=f(x1)
Then [ x 1 , x 2 , … , x n + 1 ; f ] x 1 , x 2 , … , x n + 1 ; f [x_(1),x_(2),dots,x_(n+1);f]\left[\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{\mathrm{n}+1} ; \mathrm{f}\right][x1,x2,…,xn+1;f]the gradient n n nnn-th order in the n + 1 n + 1 n+1n+1n+1points x 1 , x 2 , … , x n + 1 x 1 , x 2 , … , x n + 1 x_(1),x_(2),dots,x_(n+1)x_{1}, x_{2}, \ldots, x_{n+1}x1,x2,…,xn+1(which are assumed to be different from each other).
A function f f fffare called non-concave, convex, concave or non-convex n n nnn-th order, if the gradients ( n + 1 n + 1 n+1n+1n+1)-th order [ x 1 , x 2 , … , x n + 2 ; f ] x 1 , x 2 , … , x n + 2 ; f [x_(1),x_(2),dots,x_(n+2);f]\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right][x1,x2,…,xn+2;f]for each group of n + 2 n + 2 n+2n+2n+2various nodes x 1 , x 2 , … , x n + 2 , m ≧ , > , < x 1 , x 2 , … , x n + 2 , m ≧ , > , < x_(1),x_(2),dots,x_(n+2),m >= , > , <\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{\mathrm{n}+2}, \mathrm{~m} \geqq,>,<x1,x2,…,xn+2, m≧,>,<or ≦ 0 ≦ 0 <= 0\leqq 0≦0In particular, you will receive for n = − 1 n = − 1 n=-1n=-1n=−1the functions with invariant sign, non-negative, positive, negative or non-positive functions and for n = 0 n = 0 n=0\mathrm{n}=0n=0the monotonic functions, non-decreasing, increasing, decreasing and non-increasing functions. For n = 1 n = 1 n=1n=1n=1we have the usual non-concave, convex, concave, and non-convex functions, respectively.
8. For each function f f fffthere are polynomials P P PPPn-th degree, so that the equation f ( x ) = P ( x ) f ( x ) = P ( x ) f(x)=P(x)f(x)=P(x)f(x)=P(x)in at least n + 1 n + 1 n+1n+1n+1different points. To prove this, it is sufficient n + 1 n + 1 n+1n+1n+1different points x 1 x 1 x_(1)x_{1}x1, x 2 , … , x n + 1 x 2 , … , x n + 1 x_(2),dots,x_(n+1)x_{2}, \ldots, x_{n+1}x2,…,xn+1from the domain of the function f f fffand the Lagrange interpolation polynomial with respect to the function f f fffin the junctions x α , α = 1 , 2 , … , n + 1 x α , α = 1 , 2 , … , n + 1 x_(alpha),alpha=1,2,dots,n+1x_{\alpha}, \alpha=1,2, \ldots, n+1xa,a=1,2,…,n+1to consider. If P P PPPthis polynomial is, then it is n n nnn-th degree, and we have f ( x α ) = P ( x α ) , α = 1 , 2 , … , n + 1 f x α = P x α , α = 1 , 2 , … , n + 1 f(x_(alpha))=P(x_(alpha)),alpha=1,2,dots,n+1f\left(x_{\alpha}\right)=P\left(x_{\alpha}\right), \alpha=1,2, \ldots, n+1f(xa)=P(xa),a=1,2,…,n+1.
The functions f f fff, for which the equation f ( x ) = P ( x ) f ( x ) = P ( x ) f(x)=P(x)f(x)=P(x)f(x)=P(x)at most n + 1 n + 1 n+1n+1n+1Zeros for each polynomial P P PPPn-th degree, the single-valued functions generalize (the property applies to n = 0 n = 0 n=0n=0n=0. Then we have the following generalization of Theorem 1:
Theorem 3. In order for the function defined in interval I and continuous f f fffconvex or concave of nth order, it is necessary and sufficient that the equation f ( x ) = P ( x ) f ( x ) = P ( x ) f(x)=P(x)f(x)=P(x)f(x)=P(x)for every polynomial of degree n at most n + 1 n + 1 n+1\mathrm{n}+1n+1has zeros.
The condition is necessary if there is a polynomial P of n-th degree such that f ( x α ) = P ( x α ) f x α = P x α f(x_(alpha))=P(x_(alpha))f\left(x_{\alpha}\right)=P\left(x_{\alpha}\right)f(xa)=P(xa), α = 1 , 2 , … , n + 2 α = 1 , 2 , … , n + 2 alpha=1,2,dots,n+2\alpha=1,2, \ldots, n+2a=1,2,…,n+2, where the points x α x α x_(alpha)x_{\alpha}xaare different from each other, we would have [ x 1 , x 2 , … , x n + 2 ; f ] = 0 x 1 , x 2 , … , x n + 2 ; f = 0 [x_(1),x_(2),dots,x_(n+2);f]=0\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right]=0[x1,x2,…,xn+2;f]=0, and the function could be non-convex or concave of nth order.
The condition is also sufficient. Assuming the condition in Theorem 3 is met, two cases can occur.
  1. There are n + 2 n + 2 n+2n+2n+2different points x 1 , x 2 , … , x n + 2 ∈ I x 1 , x 2 , … , x n + 2 ∈ I x_(1),x_(2),dots,x_(n+2)in Ix_{1}, x_{2}, \ldots, x_{n+2} \in Ix1,x2,…,xn+2∈I, so that [ x 1 , x 2 , … , x n + 2 ; f ] = 0 x 1 , x 2 , … , x n + 2 ; f = 0 [x_(1),x_(2),dots,x_(n+2);f]=0\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right]=0[x1,x2,…,xn+2;f]=0In this case, the conditions P ( x α ) = f ( x α ) , α = 1 , 2 , … , n + 1 P x α = f x α , α = 1 , 2 , … , n + 1 P(x_(alpha))=f(x_(alpha)),alpha=1,2,dots,n+1P\left(x_{\alpha}\right)=f\left(x_{\alpha}\right), \alpha=1,2, \ldots, n+1P(xa)=f(xa),a=1,2,…,n+1certain polynomial P of n-th degree also the condition P ( x n + 2 ) = f ( x n + 2 ) P x n + 2 = f x n + 2 P(x_(n+2))=f(x_(n+2))P\left(x_{n+2}\right)=f\left(x_{n+2}\right)P(xn+2)=f(xn+2), which contradicts the assumption.
  2. There are different points x 1 ′ , x 2 ′ , … , x n + 2 ′ ∈ I x 1 ′ , x 2 ′ , … , x n + 2 ′ ∈ I x_(1)^('),x_(2)^('),dots,x_(n+2)^(')in Ix_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+2}^{\prime} \in Ix1′,x2′,…,xn+2′∈Iand the different points x 1 ′ ′ , x 2 ′ ′ , … , x n + 2 ′ ′ ∈ I x 1 ′ ′ , x 2 ′ ′ , … , x n + 2 ′ ′ ∈ I x_(1)^(''),x_(2)^(''),dots,x_(n+2)^('')in Ix_{1}^{\prime \prime}, x_{2}^{\prime \prime}, \ldots, x_{n+2}^{\prime \prime} \in Ix1′′,x2′′,…,xn+2′′∈I, so that [ x 1 ′ , x 2 ′ , … , x n + 2 ′ ; f ] = A < 0 , [ x 1 ′ ′ , x 2 ′ ′ , … , x n + 2 ′ ′ ; f ] = B > 0 x 1 ′ , x 2 ′ , … , x n + 2 ′ ; f = A < 0 , x 1 ′ ′ , x 2 ′ ′ , … , x n + 2 ′ ′ ; f = B > 0 [x_(1)^('),x_(2)^('),dots,x_(n+2)^(');f]=A < 0,[x_(1)^(''),x_(2)^(''),dots,x_(n+2)^('');f]=B > 0\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+2}^{\prime} ; f\right]=A<0,\left[x_{1}^{\prime \prime}, x_{2}^{\prime \prime}, \ldots, x_{n+2}^{\prime \prime} ; f\right]=B>0[x1′,x2′,…,xn+2′;f]=A<0,[x1′′,x2′′,…,xn+2′′;f]=B>0. If we assume that x 1 ′ < x 2 ′ < … < x n + 2 ′ , x 1 ′ ′ < x 2 ′ ′ < … < x n + 2 ′ ′ x 1 ′ < x 2 ′ < … < x n + 2 ′ , x 1 ′ ′ < x 2 ′ ′ < … < x n + 2 ′ ′ x_(1)^(') < x_(2)^(') < dots < x_(n+2)^('),x_(1)^('') < x_(2)^('') < dots < x_(n+2)^('')x_{1}^{\prime}<x_{2}^{\prime}<\ldots<x_{n+2}^{\prime}, x_{1}^{\prime \prime}<x_{2}^{\prime \prime}<\ldots<x_{n+2}^{\prime \prime}x1′<x2′<…<xn+2′,x1′′<x2′′<…<xn+2′′and x α = λ x α ′ + ( 1 − λ ) x α ′ ′ α = 1 , 2 , … , n + 2 x α = λ x α ′ + ( 1 − λ ) x α ′ ′ α = 1 , 2 , … , n + 2 x_(alpha)=lambdax_(alpha)^(')+(1-lambda)x_(alpha)^('')alpha=1,2,dots,n+2x_{\alpha}=\lambda x_{\alpha}^{\prime}+(1-\lambda) x_{\alpha}^{\prime \prime} \alpha=1,2, \ldots, n+2xa=lxa′+(1−l)xa′′a=1,2,…,n+2take, you can see just as in the case n = 0 n = 0 n=0n=0n=0of sentence 1 that it is a λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)l∈(0,1)for which [ x 1 , x 2 , … , x n + 2 ; f ] = 0 x 1 , x 2 , … , x n + 2 ; f = 0 [x_(1),x_(2),dots,x_(n+2);f]=0\left[\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{\mathrm{n}+2} ; \mathrm{f}\right]=0[x1,x2,…,xn+2;f]=0. Thus we return to Case 1. The continuity of the gradient
    [ x 1 , x 2 , … , x n + 2 ; f ] x 1 , x 2 , … , x n + 2 ; f [x_(1),x_(2),dots,x_(n+2);f]\left[\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{\mathrm{n}+2} ; \mathrm{f}\right][x1,x2,…,xn+2;f]for λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]l∈[0,1]follows from its spelling in the following form
    | 1 x 1 x 1 2 … x 1 n f ( x 1 ) 1 x 2 x 2 2 … x 2 n f ( x 2 ) 1 x n + 2 ⋯ x n + 2 2 f ( x n + 2 ) | 1      x 1      x 1 2 … x 1 n      f x 1 1      x 2      x 2 2 … x 2 n      f x 2 1      x n + 2      ⋯      x n + 2 2      f x n + 2 |[[1,x_(1),x_(1)^(2)dotsx_(1)^(n),f(x_(1))],[1,x_(2),x_(2)^(2)dotsx_(2)^(n),f(x_(2))],[1,x_(n+2),cdots,x_(n+2)^(2),f(x_(n+2))]]|\left|\begin{array}{|lllll|}1 & x_{1} & x_{1}^{2} \ldots x_{1}^{n} & f\left(x_{1}\right) \\ 1 & x_{2} & x_{2}^{2} \ldots x_{2}^{n} & f\left(x_{2}\right) \\ 1 & x_{n+2} & \cdots & x_{n+2}^{2} & f\left(x_{n+2}\right)\end{array}\right||1x1x12…x1nf(x1)1x2x22…x2nf(x2)1xn+2⋯xn+22f(xn+2)|and from the fact that we x 1 < x 2 < … < x n + 2 x 1 < x 2 < … < x n + 2 x_(1) < x_(2) < dots < x_(n+2)\mathrm{x}_{1}<\mathrm{x}_{2}<\ldots<\mathrm{x}_{\mathrm{n}+2}x1<x2<…<xn+2have if λ ϵ [ 0 , 1 ] λ ϵ [ 0 , 1 ] lambda epsilon[0,1]\lambda \epsilon[0,1]lϵ[0,1].
This proves Theorem 3.
9. Another class of functions that had to be investigated is the continuous functions f f fffformed, which enjoy the property that the equation f ( x ) = P ( x ) f ( x ) = P ( x ) f(x)=P(x)f(x)=P(x)f(x)=P(x)for any polynomial n n nnn-th degree at most n + 2 n + 2 n+2n+2n+2, or more generally n + k n + k n+kn+kn+k ( k k kkkis a given natural number) has zeros. It would be interesting to see what relationship exists between these functions and the nth-order functions on segments that we introduced in the work //, which also characterize certain behaviors more general than higher-order convexity.
    1. Prof., Dr., Cluj, Academy of the Socialist Republic of Romania, Cluj Branch, Institute of Computing
  2. This function is three-valued and increasing in each of the intervals and decreasing in each of the intervals
    1. Dr., Moscow, Academy of Sciences of the USSR, Computing Center
1967

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