In Proposition 3, why would the definition of Hölder’s inequality requires the exponents r to satisfy this formula 1/r = 1/p+1/q? To me, this is a bit bizzard and I don’t know the reason to do so.

[See some previous discussion at https://terrytao.wordpress.com/2008/12/27/tricks-wiki-use-basic-examples-to-calibrate-exponents/ -T.]

Also, in 5th line after equation (29) should say Ric is a (0,2) tensor.

[Corrected, thanks – T.]

[Corrected, thanks – T.]

We have localized to in a previous step, so we are already effectively working in a bounded annulus.

Dear Professor Tao, Thank you for the clarification on Theorem 8.

Having reread your proof of Theorem 7 carefully, I have a follow-up question about whether the same dominated convergence argument extends to Theorem 8. In the proof of Theorem 7, the dominated convergence argument handles the “middle annulus” cleanly. After establishing the Gaussian domination for , the remaining region corresponds under to the fixed compact set , independent of . Continuity of and the nonlattice condition give a uniform on this set, so is dominated by for some independent of .

For Theorem 8, the integration domain is all of , and the analogous middle annulus corresponds to . Of course, the integrand carries an additional factor where is a fixed Schwartz function independent of . However, for fixed and , we have , a nonzero constant, so the Schwartz decay in degrades as grows.

Does this Schwartz factor nonetheless resolve the domination issue? Or I wonder if additional input needed for the unbounded annulus, analogous to what Stone (1965) achieves by introducing a kernel with compactly supported Fourier transform , which eliminates the unbounded region entirely, with the remaining middle region estimate deferred to Rva\v{c}eva (1954)?

Thank you again for your time!

PS: As a minor note, I think the key expression at the end of the proof of Theorem 8 should read for consistency.

]]> https://terrytao.wordpress.com/books/analysis-i/comment-page-17/#comment-692275 Sun, 19 Apr 2026 15:53:38 +0000 http://terrytao.wordpress.com//analysis-i/#comment-692275 In reply to Terence Tao.

Thanks for your encouragement!

According to your textbook, when a function is discontinuous, there are four types of discontinuities: jump, removable, asymptotic, and oscillating discontinuities. I tested the Darboux theorem one by one with examples for each type and found that the first three types did not satisfy it. Only when I got to the fourth type—oscillating discontinuity—did I find that it actually satisfied the theorem?! The example I used was x^{2}sin(1/x).

I would like to ask: what is the deeper reason behind this? Why do functions with the first three types of discontinuities fail to have the intermediate value property, while those with an oscillating discontinuity can satisfy it? More generally, what are the characteristics of functions that satisfy the intermediate value property?

[Corrected, thanks – T.]

Congratulations, you have discovered Darboux’s theorem! It illustrates that the intermediate value theorem property is a strictly weaker property that continuity; continuous functions obey the intermediate value theorem, but they are not the only class of functions to do so.

I discuss various aspects of this in this recent podcast interview (particularly near the end). I think it is inevitable that mathematical practice will have to bifurcate somewhat into certain types of activities (e.g., routine proofs and calculations, or scanning large sets of problems for “quick wins”) that are currently done by humans but will soon be primarily offloaded to AI (except for student training purposes, or certain types of competitions), and other types of activities (such as trying to coherently digest and construct narratives for a body of literature, or to assess the promise of some recently discovered technique or concept) that we would want to keep primarily done by humans. In particular, our profession will need to determine which portions of our work should prioritize efficiency, and which portions should prioritize other goals; this has not been a major concern in the past because gains in efficiency have generally been correlated with gains in other objectives such as gaining insight or training the next generation of mathematicians, but AI is beginning to decouple these objectives from each other.

I believe there will be several policy guidelines coming out soon by various groups of mathematicians or professional societies that will give more guidance in this regard.

Besides the vanilla extract analogy, the other main advice we give in the paper is to make a “blue team”/”red team” distinction, and restrict AI use to either (a) red-teaming one’s own work (e.g., using AI to proofread your human-generated text), or (b) blue-team work that is within the capability of you (and the tools you know how to use) to “red team”. Examples of the latter include:

• Using AI to do literature search when you are capable of following the provided references and checking that they do indeed establish what the AI claims they do;
• Using AI to provide the history of a mathematical concept or terminology, if you are capable of double-checking the claims through non-AI methods (e.g., web search);
• Using AI to generate code for some numerical task (e.g., plotting a function) or formalization task (e.g., proving something in Lean), provided that you understand the programming language well enough to read the code, verify it for correctness, and modify or debug it as necessary;
• Using AI to format text, if you are capable of checking that the reformatting is proceeding as you intended it to; or
• Using AI to perform a calculation, if you already know how to perform the calculation and can make various “sanity checks” to ensure that the answer is correct, or at least consistent with everything else you know around the calculation.

More risky would be the practice of asking an AI for possible proof strategies for a problem that you do not know how to solve. If you are stuck and desperately in need of a solution with no other resources available than AI assistance, I would recommend instead a red-team approach: posing your own proof strategies to such an AI model and asking it to critique them for strengths and weaknesses. (You may need to explicitly prompt such an AI not to provide you directly with a solution, and state that your primary goal is to develop your skills and intuition rather than simply obtain an answer to the question.)

The use case to avoid if at all possible is simply to ask the AI for the answer to a problem that you are unable to solve on your own; even setting aside the problem that the answer may be incorrect, give off the “tells” of AI-generated writing, or utilize techniques that you are not familiar with, a reliance on such practices will compromise your ability to solve such problems by yourself in the future. As a rule of thumb: if you would be unable to coherently present the output of the AI in a class presentation and be able to answer questions about it without further AI assistance, it should not be part of your workflow.