108 1. Measure theory it is quite a weak notion of convergence, lacking many of the properties of the modes of convergence discussed here. 1.6. Differentiation theorems Let [a, b] be a compact interval of positive length (thus −∞ a b +∞). Recall that a function F : [a, b] → R is said to be differentiable at a point x ∈ [a, b] if the limit (1.19) F (x) := lim y→x y∈[a,b]\{x} F (y) − F (x) y − x exists. In that case, we call F (x) the strong derivative, classical derivative, or just derivative for short, of F at x. We say that F is everywhere differen- tiable, or differentiable for short, if it is differentiable at all points x ∈ [a, b], and differentiable almost everywhere if it is differentiable at almost every point x ∈ [a, b]. If F is differentiable everywhere and its derivative F is continuous, then we say that F is continuously differentiable. Remark 1.6.1. In §1.13 of An epsilon of room, Vol. I, the notion of a weak derivative or distributional derivative is introduced. This type of derivative can be applied to a much rougher class of functions and is in many ways more suitable than the classical derivative for doing “Lebesgue” type anal- ysis (i.e. analysis centered around the Lebesgue integral, and in particular, allowing functions to be uncontrolled, infinite, or even undefined on sets of measure zero). However, for now we will stick with the classical approach to differentiation. Exercise 1.6.1. If F : [a, b] → R is everywhere differentiable, show that F is continuous and F is measurable. If F is almost everywhere differentiable, show that the (almost everywhere defined) function F is measurable (i.e. it is equal to an everywhere defined measurable function on [a, b] outside of a null set), but give an example to demonstrate that F need not be continuous. Exercise 1.6.2. Give an example of a function F : [a, b] → R which is ev- erywhere differentiable, but not continuously differentiable. (Hint: Choose an F that vanishes quickly at some point, say at the origin 0, but which also oscillates rapidly near that point.) In single-variable calculus, the operations of integration and differenti- ation are connected by a number of basic theorems, starting with Rolle’s theorem. Theorem 1.6.2 (Rolle’s theorem). Let [a, b] be a compact interval of pos- itive length, and let F : [a, b] → R be a differentiable function such that F (a) = F (b). Then there exists x ∈ (a, b) such that F (x) = 0.

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