Intermediate value theorem

Intermediate value theorem Intermediate value theorem: Let {displaystyle f} be a continuous function defined on {displaystyle [a,b]} and let {displaystyle s} be a number with {displaystyle f(a)0} . Since {displaystyle f} is continuous, there is a {displaystyle delta >0} such that {displaystyle |f(x)-f(c)|f(a^{**})-varepsilon >u-varepsilon .} Both inequalities {displaystyle u-varepsilon 0} , from which we deduce {displaystyle f(c)=u} as the only possible value, as stated.

Remark: The intermediate value theorem can also be proved using the methods of non-standard analysis, which places "intuitive" arguments involving infinitesimals on a rigorous footing.[4] History A form of the theorem was postulated as early as the 5th century BCE, in the work of Bryson of Heraclea on squaring the circle. Bryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area.[5] The theorem was first proved by Bernard Bolzano in 1817. Bolzano used the following formulation of the theorem:[6] Let {displaystyle f,phi } be continuous functions on the interval between {displaystyle alpha } and {displaystyle beta } such that {displaystyle f(alpha )phi (beta )} . Then there is an {displaystyle x} between {displaystyle alpha } and {displaystyle beta } such that {displaystyle f(x)=phi (x)} .

The equivalence between this formulation and the modern one can be shown by setting {displaystyle phi } to the appropriate constant function. Augustin-Louis Cauchy provided the modern formulation and a proof in 1821.[7] Both were inspired by the goal of formalizing the analysis of functions and the work of Joseph-Louis Lagrange. The idea that continuous functions possess the intermediate value property has an earlier origin. Simon Stevin proved the intermediate value theorem for polynomials (using a cubic as an example) by providing an algorithm for constructing the decimal expansion of the solution. The algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration.[8] Before the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function. Proponents include Louis Arbogast, who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.[9] Earlier authors held the result to be intuitively obvious and requiring no proof. The insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of infinitesimals in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.

Generalizations The intermediate value theorem is closely linked to the topological notion of connectedness and follows from the basic properties of connected sets in metric spaces and connected subsets of R in particular: If {displaystyle X} and {displaystyle Y} are metric spaces, {displaystyle fcolon Xto Y} is a continuous map, and {displaystyle Esubset X} is a connected subset, then {displaystyle f(E)} is connected. (*) A subset {displaystyle Esubset mathbb {R} } is connected if and only if it satisfies the following property: {displaystyle x,yin E, x 0 and f(0) = 0. This function is not continuous at x = 0 because the limit of f(x) as x tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by the Conway base 13 function.

In fact, Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property (even though they need not be continuous).

Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions;[11] this definition was not adopted.

In constructive mathematics In constructive mathematics, the intermediate value theorem is not true. Instead, one has to weaken the conclusion: Let {displaystyle a} and {displaystyle b} be real numbers and {displaystyle f:[a,b]to R} be a pointwise continuous function from the closed interval {displaystyle [a,b]} to the real line, and suppose that {displaystyle f(a)<0} and {displaystyle 00} there exists a point {displaystyle x} in the unit interval such that {displaystyle vert f(x)vert

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