Timeless Theorems of Mathematics/Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental theorem in calculus. The theorem states that if a function, is continuous on a closed interval then for any value, defined between and there exists at least one value such that . thumb|Intermediate Value Theorem: f(x) is continuous on [a, b], there exists at least one value c, that is defined on (a, b) such that f(c) = y. Proof Statement: If a function, is continuous on then for every between and there exists at least one value such that Proof: Assume that is a continuous function on and Consider a function The purpose of defining is to investigate the behavior of concerning the value . Since is continuous on and is a constant, is also continuous on as the difference of two continuous functions is continuous. Now, [As and ] Or, In the same way, Since is continuous and is defined below the -axis while is defined above the -axis, there must exist at least one point in the interval where . Therefore, at the point , ∴ There exists at least one point in the interval such that [Proved]