Counterexamples in analysis
by Bernard R. Gelbaum · first published 1964
79% recommend21% neutral
Why HN recommends it
Collection of pathological examples that corrects calculus-based intuitions about continuity, differentiability and integrability. Suited as a companion for beginners in analysis and proof-based courses.
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In addition to everything else you do, I recommend the following book: "Counterexamples in Analysis" by Gelbaum and Olmsted. You will find that many of your intuitions you picked up in calculus are violated in analysis. For instance, in calculus, many examples are both continuous and differentiable…
on “Ask HN: Resources to learn real analysis?” ↗…conditions in optimization. Or, any closed set can be the level set of an infinitely differentiable function. Sure, really fun reading for such things is: Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964.
on “How to Learn Advanced Mathematics Without Heading to University” ↗…elegant, but maybe don't work way too hard on the exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles. Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability…
on “Ask HN: Older textbooks/papers you consider classics still worth studying today?” ↗…take a higher level topic(a intro to analysis or abstract algebra) to get a feel of how to apply these techniques in the intro proof class. It's a long process and there is no easy way. See this book, Counterexamples in analysis. http://www.amazon.com/Counterexamples-Analysis-Dover-Books-M... Half of the things I would believe to be true from intuitive argument turns out to be completely wrong. Finally, one might question why…
on “Math Intuition Cheatsheet” ↗…what learn via the Baire category theorem can shake one's intuitive view of the real line and our 3-space. Definitely a masterpiece. Maybe it's profound. Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis When studying Rudin, Royden, etc., don't be without this one! And it's astounding and clears up a lot. Or, why didn't Rudin state the theorem this way? Because that way it's not true -- see Gelbaum…
on “What's a Mathematician to do?” ↗…continuous everywhere except on a set of measure zero. Nice. Now an exercise is to find a function that is differentiable but whose derivative is not Riemann integrable. Might look in Gelbaum and Olmstead, Counterexamples in Analysis. Also the later editions of baby Rudin cover the exterior algebra of differential forms. That material is of interest in differential geometry, some applications, and general relativity. That's an…
on “Confession as an AI researcher; seeking advice” ↗…of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \ An amazing weekend. Keep at hand Bernard R.\ Gelbaum and John M.\ H.\ Olmsted, {\it Counterexamples in Analysis,\/} Holden-Day, San Francisco, 1964.\ \ Notice that for the Lebesgue integral, there is one, central proof technique in just four steps: Prove the theorem for a function that is just a positive constant defined…
on “Ask HN: Resources to learn real analysis?” ↗For anyone looking for more of these, you may enjoy the book Counterexamples in Analysis (http://www.amazon.com/Counterexamples-Analysis-Dover-Books-M...)
on “Weierstrass functions: continuous but not differentiable anywhere” ↗…Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious. Counterexamples In Analysis https://www.amazon.com/Counterexamples-Analysis-Dover-Books-... (Pdf version) https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...
on “Calculus Is Deeply Irrational” ↗Gelbaum-Olmsted Counterexamples in analysis is a standard reference for various pathological examples in analysis: https://archive.org/details/counterexamplesi0000gelb
on “Notes on a few abnormalities in analysis (2022)” ↗A favorite on my bookshelf: Bernard R.\ Gelbaum and John M.\ H.\ Olmsted, {\it Counterexamples in Analysis,\/} Holden-Day, San Francisco, 1964.\ \
on “Weierstrass's Monster” ↗…& Hubbard for a beautiful and beginner-friendly mix of algebra and analysis. My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.
on “Mathematicians are chronically lost and confused (2014)” ↗…heard of it. Hardly anyone has actually read it. For the math books I'm not going to recommend standard text books much. My recommendations are going to be books such as Counterexamples in Topology, Counterexamples in Analysis. Those are super fun books to work through as counterexamples warp your mind a little bit, and you get that tiny adrenaline rush when you get it. Another example is To Mock a Mockingbird. It's not a text…
on “My Top 100 Programming, Computer and Science Books: Part One” ↗No, it's not standard. I doubt that it is in Counterexamples in Analysis, one of my favorite books. No, don't need a non-empty interior for the closed set. The closed set can even be empty. The function is to be zero on the closed set and positive otherwise. For a single…
on “Kenneth Arrow and the golden age of economic theory” ↗Analysis in general is notorious for these kinds of things. There's a neat little Dover book called Counterexamples in Analysis full of them.
on “How a destructive idea paved the way for modern math” ↗