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Intermediate Value Theorem
Transcript of Intermediate Value Theorem
1781-1848 University of Prague
1804 Bolzano became a priest, and was appointed to the chair of philosophy of religion in Prague. Philanthropy Oppression in Austrian Empire
repressive political system
Bolzano's ideas threatened Emporer Franz Bolzano wrote of logic and theory, which generally reflected his liberal viewpoints. It is said that someone who was not found of Bolzano pointed out one of his passages to Emporer Franz:
"There will come a time when the thousand different kinds of orders of rank and barriers among human beings will be put back into their proper confines, when everyone will deal with his neighbor as a brother." 1820: Bolzano was exiled to the Austrian countryside where he was prohibited from teaching and publishing any of his work. Bolzano=Optimist
more time to spend on mathematics and logic Logic A propositional form F is logically valid iff F is a logical propositional form that is universally valid, i.e., at least one of its members is true, and all of its members with non-empty subject ideas are true. F is logically contravalid iff F is a logical propositional form, and all of its members are false. Geometry Foundation:
wrote a book called "Anti-Euklid," but this did not contradict Euclid, only enhanced ideas Work with infinitesimal numbers:
In his book, "Pure Theory of Numbers," he provided for the first time a precision of the concept of an infinitely small and infinitely great number While he was in exile, he discovered the first analytic proof to the intermediate value theorem. However, the mathematician, Cauchy, also came up with a proof to the IVT, but when they discovered Bolzano's work, they gave him credit as well. IVT:
If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c [a, b] such that f(c) = u Proof of Bolzano's Theorem!
Case 1: Say f(c ) is less than 0, so c is a member of S. By the Sign-Preserving Property of Continuous Functions, the existence of an open interval containing c where f takes only negative values thus contains points greater than c. So, c cannot be an upper bound for S. This is a contradiction.
Case 2:Suppose now f is more than 0. Similarly, since f is continuous there is an open interval constraint c where f is positive. But this interval contains points less than c, which must be upper bounds for S. But, this contradicts the assumption that c is a least upper bound for S, so f(c ) cannot be greater than 0.Case 3: Because Case 1 and Case 2 fail, f(c ) must equal 0. Bolzano then went on to look at the case of when u=0 and proved this.
This proof became known as the Bolzano Theorem.