The Intermediate Value Theorem is a theorem in calculus that states that if a continuous function has a different sign at two points in its domain, then it must have at least one root (or zero) between those two points.
The Intermediate Value Theorem is important because it guarantees the existence of roots for certain continuous functions. This allows us to use the theorem to prove the existence of solutions to equations that cannot be solved algebraically.
The Intermediate Value Theorem is used in a variety of real-life applications, such as in economics to prove the existence of equilibrium points, in physics to prove the existence of solutions to certain equations, and in computer science to design algorithms for finding roots of functions.
The Intermediate Value Theorem requires that the function is continuous on a closed interval and that the function has a different sign at the endpoints of the interval. If these conditions are met, then the theorem guarantees the existence of at least one root on that interval.
No, the Intermediate Value Theorem only applies to continuous functions. If a function is not continuous, then it is not guaranteed to have a root between two points where it has a different sign. However, there are other theorems and methods that can be used to find roots of non-continuous functions.