Converse of the Intermediate Value Theorem
The converse of the Intermediate Value Theorem (IVT) is not always true. The converse statement is stated as follows:
If there exists a point c ∈ [a, b] such that f(c)=d for every number d between f(a) and f(b), then f is continuous for the interval [a, b].
The above statement is not true always. A function can follow the Intermediate Value Theorem despite being discontinuous. In other words, a function following the IVT property need not to be a continuous function but a continuous function always follows the Intermediate Value Theorem.
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Intermediate Value Theorem
Intermediate Value Theorem is a theorem in calculus which defines an important property of continuous functions. It is abbreviated as IVT. The theorem is quite intuitive one but provides a significant result for the interpretation of the behaviour of functions. It can be used to know the range of values for a physical quantity such as temperature if an expression in terms of time or other variables is known for it. Other applications of the theorem include solving equations, proving the existence of roots, and analyzing real-world problems where continuity is observed.
In this article, we will learn the statement of the theorem, its proof by two different approaches, its various applications, the converse of the theorem, some numerical problems and related frequently asked questions.