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I would like to ask whether there is a reference which collects pathological examples in mathematics (in general). What I mean is that, for instance, consider Weierstrass function. It has the property that it is continuous everywhere but differentiable nowhere. Similarly, Cantor set has uncountably many elements but its Lebesgue measure is zero.

I know that sets and functions are different concepts but the critical progresses are generally based on these pathological examples. I wonder if there are references that completely focus on these type of examples.

Also, I am sure that there are many interesting functions, sets or other objects that teach much about the related subjects. Other reference suggestions, which examine the subjects over these examples, in that sense are more than welcome.

Thanks!

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I have a book sitting on my shelf called "Counterexamples in topology". It tends to have a lot of these nasty examples –  muzzlator Mar 3 '13 at 21:14
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@muzzlator: Nice examples! :-) There’s also Gelbaum & Olmsted, Counterexamples in Analysis. –  Brian M. Scott Mar 3 '13 at 21:16
    
the topology: store.doverpublications.com/048668735x.html By now there ought to be counterexample/example books in other fields, but they may not have inexpensive reprints available. However, when I searched for counterexample on Dover, it showed five books: doverpublications.ecomm-search.com/… –  Will Jagy Mar 3 '13 at 21:19
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There is an electronic version of the database from Steen and Seebach's "Counterexamples in Topology" called Spacebook. –  Austin Mohr Mar 17 '13 at 21:34
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up vote 8 down vote accepted

This is perhaps useful. I don't know if the price was a constraint or not because in the comments Dover Publications are emphasized. Both counterexamples in analysis and topology are pretty good but there are more.

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Good. Once you know correct title, author(s), and so on, you can check for used at both abebooks.com and bookfinder.com with occasional great bargains. Hit and miss on price. –  Will Jagy Mar 3 '13 at 22:33
    
That's exactly how I got through college. –  Fixed Point Mar 4 '13 at 0:14
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