除了你手头在搞的所有东西之外，我还推荐下面这本书：Gelbaum 和 Olmsted 写的《Counterexamples in Analysis》（分析中的反例）。你会发现，你在微积分里学到的很多直觉在数学分析里全都被推翻了。比如，在微积分里，很多例子既连续又可导……
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…
……最优化中的条件。或者，任何闭集都可以是某个无穷次可微函数的水平集。当然，读这类东西特别有意思的书是：Bernard R. Gelbaum 和 John M. H. Olmsted 的《Counterexamples in Analysis》，Holden-Day 出版，旧金山，1964年。
…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.
……很优雅，但也许没必要在关于上/下半连续性的习题上死磕太久，而且有个总结比 Littlewood 的三大原则更好。Bernard R. Gelbaum 和 John M. H. Olmsted 的《Counterexamples in Analysis》；John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》；Walter Rudin 的《Real and Complex Analysis》；Walter Rudin 的《Functional Analysis》；Leo Breiman 的《Probability》……
…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…
……去上一门更高阶的课（比如分析导论或抽象代数），去感受一下在入门证明课里怎么应用这些技巧。这是一个漫长的过程，没有任何捷径。看看这本书，《Counterexamples in Analysis》。http://www.amazon.com/Counterexamples-Analysis-Dover-Books-M... 我凭直觉以为是对的东西，有一半最后都被证明错得一塌糊涂。最后，有人可能会问为什么……
…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…
……通过贝尔纲定理学到的东西，足以颠覆一个人对实数线和我们所在的三维空间的直觉认知。绝对是一部杰作。甚至可以说很深刻。Bernard R. Gelbaum 和 John M. H. Olmsted 的《Counterexamples in Analysis》。在学 Rudin、Royden 等人的教材时，手头绝不能少了这本！它让人眼界大开，能澄清很多困惑。或者，为什么 Rudin 没有那样表述那个定理？因为要是那样表述它就不成立了——参见 Gelbaum……
…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…
……除了在测度为零的集合上之外处处连续。很漂亮。现在有一道习题是找一个可微但其导数不可黎曼积分的函数。可以翻翻 Gelbaum 和 Olmsted 的《Counterexamples in Analysis》。另外，baby Rudin 的后续版本涵盖了微分形式的外代数。那部分内容在微分几何、某些应用以及广义相对论中都很有用。那是一本……
…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…
除了你手头在搞的所有东西之外，我还推荐下面这本书：Gelbaum 和 Olmsted 写的《Counterexamples in Analysis》（分析中的反例）。你会发现，你在微积分里学到的很多直觉在数学分析里全都被推翻了。比如，在微积分里，很多例子既连续又可导……
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…
……最优化中的条件。或者，任何闭集都可以是某个无穷次可微函数的水平集。当然，读这类东西特别有意思的书是：Bernard R. Gelbaum 和 John M. H. Olmsted 的《Counterexamples in Analysis》，Holden-Day 出版，旧金山，1964年。
…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.
……很优雅，但也许没必要在关于上/下半连续性的习题上死磕太久，而且有个总结比 Littlewood 的三大原则更好。Bernard R. Gelbaum 和 John M. H. Olmsted 的《Counterexamples in Analysis》；John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》；Walter Rudin 的《Real and Complex Analysis》；Walter Rudin 的《Functional Analysis》；Leo Breiman 的《Probability》……
…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…
……去上一门更高阶的课（比如分析导论或抽象代数），去感受一下在入门证明课里怎么应用这些技巧。这是一个漫长的过程，没有任何捷径。看看这本书，《Counterexamples in Analysis》。http://www.amazon.com/Counterexamples-Analysis-Dover-Books-M... 我凭直觉以为是对的东西，有一半最后都被证明错得一塌糊涂。最后，有人可能会问为什么……
…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…
……通过贝尔纲定理学到的东西，足以颠覆一个人对实数线和我们所在的三维空间的直觉认知。绝对是一部杰作。甚至可以说很深刻。Bernard R. Gelbaum 和 John M. H. Olmsted 的《Counterexamples in Analysis》。在学 Rudin、Royden 等人的教材时，手头绝不能少了这本！它让人眼界大开，能澄清很多困惑。或者，为什么 Rudin 没有那样表述那个定理？因为要是那样表述它就不成立了——参见 Gelbaum……
…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…
……除了在测度为零的集合上之外处处连续。很漂亮。现在有一道习题是找一个可微但其导数不可黎曼积分的函数。可以翻翻 Gelbaum 和 Olmsted 的《Counterexamples in Analysis》。另外，baby Rudin 的后续版本涵盖了微分形式的外代数。那部分内容在微分几何、某些应用以及广义相对论中都很有用。那是一本……
…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…