1.6. Differentiation theorems 109 Proof. By subtracting a constant from F (which does not affect differen- tiability or the derivative) we may assume that F (a) = F (b) = 0. If F is identically zero, then the claim is trivial, so assume that F is non-zero somewhere. By replacing F with −F if necessary, we may assume that F is positive somewhere, thus supx∈[a,b] F (x) 0. On the other hand, as F is continuous and [a, b] is compact, F must attain its maximum somewhere, thus there exists x ∈ [a, b] such that F (x) ≥ F (y) for all y ∈ [a, b]. Then F (x) must be positive and so x cannot equal either a or b, and thus must lie in the interior. From the right limit of (1.19) we see that F (x) ≤ 0, while from the left limit we have F (x) ≥ 0. Thus F (x) = 0 and the claim follows. Remark 1.6.3. Observe that the same proof also works if F is only differ- entiable in the interior (a, b) of the interval [a, b], so long as it is continuous all the way up to the boundary of [a, b]. Exercise 1.6.3. Give an example to show that Rolle’s theorem can fail if f is merely assumed to be almost everywhere differentiable, even if one adds the additional hypothesis that f is continuous. This example illustrates that everywhere differentiability is a significantly stronger property than almost everywhere differentiability. We will see further evidence of this fact later in these notes there are many theorems that assert in their conclusion that a function is almost everywhere differentiable, but few that manage to conclude everywhere differentiability. Remark 1.6.4. It is important to note that Rolle’s theorem only works in the real scalar case when F is real-valued, as it relies heavily on the least upper bound property for the domain R. If, for instance, we consider complex-valued scalar functions F : [a, b] → C, then the theorem can fail for instance, the function F : [0, 1] → C defined by F (x) := e2πix − 1 vanishes at both endpoints and is differentiable, but its derivative F (x) = 2πie2πix is never zero. (Rolle’s theorem does imply that the real and imaginary parts of the derivative F both vanish somewhere, but the problem is that they don’t simultaneously vanish at the same point.) Similar remarks can be made about functions taking values in a finite-dimensional vector space, such as Rn. One can easily amplify Rolle’s theorem to the mean value theorem: Corollary 1.6.5 (Mean value theorem). Let [a, b] be a compact interval of positive length, and let F : [a, b] → R be a differentiable function. Then there exists x ∈ (a, b) such that F (x) = F (b)−F (a) b−a .

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