陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「245A, Notes 5: Differentiation theorems」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Let be a compact interval of positive length (thus ). Recall that a function is said to be differentiable at a point if the limit

exists. In that case, we call the strong derivative , classical derivative , or just derivative for short, of at . We say that is everywhere differentiable , or differentiable for short, if it is differentiable at all points , and differentiable almost everywhere if it is differentiable at almost every point . If is differentiable everywhere and its derivative is continuous, then we say that is continuously differentiable .

已知结果和反例

Remark 1 Much later in this sequence , when we cover the theory of distributions, 下面会 see the notion of a weak derivative or distributional derivative , which 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 analysis (i.e. analysis centred around the Lebesgue integral, and in particular allowing functions to be uncontrolled, infinite, or even undefined on sets of measure zero). Howeve

Exercise 2 If is everywhere differentiable, show that is continuous and is measurable. If is almost everywhere differentiable, show that the (almost everywhere defined) function is measurable (i.e. it is equal to an everywhere defined measurable function on outside of a null set), but give an example to demonstrate that need not be continuous.

证明或构造的主线

Exercise 3 Give an example of a function which is everywhere differentiable, but not continuously differentiable. ( Hint: choose an that vanishes quickly at some point, say at the origin , but which also oscillates rapidly near that point.)

In single-variable calculus, the operations of integration and differentiation are connected by a number of basic theorems, starting with Rolle’s theorem .

阅读时建议盯住的点

Theorem 4 (Rolle’s theorem) Let be a compact interval of positive length, and let be a differentiable function such that . Then there exists such that .

Proof: By subtracting a constant from (which does not affect differentiability or the derivative) we may assume that . If is identically zero then the claim is trivial, so assume that is non-zero somewhere. By replacing with if necessary, we may assume that is positive somewhere, thus . On the other hand, as is continuous and is compact, must attain its maximum somewhere, thus there exists such that for all . Then must be positive and so cannot equal either or , and thus must

值得单独记下的条目

• The Lebesgue differentiation theorem , which roughly speaking asserts that Corollary 14 continues to hold for almost every if is merely absolutely integrable, rather than continuous;
• A number of differentiation theorems , which assert for instance that monotone, Lipschitz, or bounded variation functions in one dimension are almost everywhere differentiable; and
• The second fundamental theorem of calculus for absolutely continuous functions.
• A quantitative estimate that upper bounds the maximal fluctuation of the linear expressions in terms of the “size” of the function (where the precise definition of “size” depends on the nature of the approximation in the first ingredient).
• Show that all measurable homomorphisms are continuous. ( Hint: for any disk centered at the origin in the complex plane, show that has positive measure for at least one , and then use the Steinhaus theorem from the previous exercise.)
• Show that is a measurable homomorphism if and only if it takes the form for all and some complex coefficients . ( Hint: first establish this for rational , and then use the previous part of this exercise.)
• For each , either , or else and .
• If does not lie in any of the intervals , then one must have for all .

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源