Rolle's Theorem
Rolle's Theorem states that if a real-valued function f(x) is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0, i.e., the derivative of the function is zero at c.
Validity When Function is Not Continuous at EndpointsNo, Rolle's Theorem does not hold when the function is not continuous at the endpoints.
The theorem's requirement of continuity on the closed interval [a, b] is crucial for its validity. If the function is not continuous at one or both of the endpoints a and b, then the intermediate value property guaranteed by continuity might not hold, and the derivative might not exist at those points.
In such cases, it's not possible to guarantee the existence of a point c in the open interval (a, b) where the derivative of the function is zero, because the conditions of the theorem are not met. The continuity of the function ensures that it is "well-behaved" on the entire interval, and this is necessary for the application of Rolle's Theorem.
In summary, Rolle's Theorem is true when the function is continuous on the closed interval and differentiable on the open interval, and it does not hold when the function is not continuous at the endpoints. The continuity of the function is essential for the theorem's validity.