Final answer:
The Mean Value Theorem states that for a function to satisfy the theorem's conditions, it must be continuous on a closed interval and differentiable on an open interval.
Step-by-step explanation:
The Mean Value Theorem states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one point c in the open interval (a,b) where the derivative of the function is equal to the average rate of change of the function over the closed interval [a,b].
By requiring the function to be continuous on the closed interval, we ensure that the function is well-behaved and does not have any sudden jumps or breaks that could disrupt the derivative. The differentiability on the open interval ensures that the derivative exists and can be calculated at every point in the interval.
For example, if we have a function that is not continuous at a certain point within the interval, it may not have a derivative at that point, which would violate the conditions of the Mean Value Theorem.