Answer:
Explanation:
You need continuity to apply the Mean Value Theorem because the Mean Value Theorem relies on connecting the start and end points of an interval with a continuous function.
A counterexample showing why continuity is required:
Consider the function f(x) defined on the interval [0, 2] as:
f(x) = {
1 if x is rational
0 if x is irrational
This function is discontinuous at every point in the interval.
Now let's try applying the MVT:
f(0) = 1
f(2) = 1
The MVT states that there should exist a c in (0,2) such that f'(c) = (f(2) - f(0))/(2-0) = 0
But this function f(x) has no derivative at any x in the interval since it is discontinuous everywhere. So the requirements for the MVT fail.
This example shows that without continuity, the Mean Value Theorem does not apply, because we cannot properly define derivatives or connect the start and end points with a continuous function. The discontinuity causes the theorem to fail.
Therefore, continuity is required to properly apply the Mean Value Theorem.