Final answer:
The Intermediate Value Theorem states that for a continuous function defined on a closed interval, if zero is between the function's values at the interval's endpoints, there is a point within that interval where the function equals zero. This theorem is essential for understanding the behavior of continuous functions and is applied in calculus and probability.
Step-by-step explanation:
The question pertains to the understanding of the Intermediate Value Theorem, which states that for a continuous function f defined on a closed interval [a,b], if 0 is between f(a) and f(b), there exists a point x=c in the interval such that f(c)=0. This theorem is applied in various fields, including calculus and probability theory, to establish that a function assumes every value between f(a) and f(b) at least once on the interval [a, b]. This is a fundamental concept in understanding how continuous functions behave.
An example application of this theorem could be in a continuous probability distribution. For instance, in the scenario that a continuous probability function is given with a certain range, say 0 ≤ x ≤ 20, we consider the probability of x being within a certain interval. The Intermediate Value Theorem can help verify the existence of a particular value within this interval. However, in continuous probability, we never say that P(x=c)=0 for any particular value c since the probability of a single point is 0.
When given two function values such as f(a) and f(b) with one being positive and the other being negative, the Intermediate Value Theorem ensures us that there is at least one root c in the interval where f(c)=0.