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Nice. Reminds me of the discrete metric in my Real Analysis course: distance is 0 if it is the same point, 1 otherwise. We used to joke that it was the perfec
by philomath_mn 2y ago
Nice.
Reminds me of the discrete metric in my Real Analysis course: distance is 0 if it is the same point, 1 otherwise.
We used to joke that it was the perfect metric for finding weird counter examples
- unkulunkulu 2y agothere is a whole book “counterexamples in analysis”, basically for each theorem they find why every condition is required. these functions and ideas based on them are indispensable. this weierstrass function is definitely a mind bender I remember from high school.
- JJMcJ 2y agoAlso "Counterexamples In Topology" - same but for topology.
- shwouchk 2y agoI don’t know if they are the same books as the names are quite generic, but i picked up both this one and the topology book sibling suggested at the library during my undergrad, and read both cover to cover. Both slim books and very clear. My favorite counterexamples in analysis were based on the cantor set as it allowed counerexamples for many different scenarios while still producing functions that were “nice” in other respects
- dgoldstein0 2y agoyup, that's a standard example / counterexample. Called the discrete metric. If you are trying to prove something about metric spaces, you should try your statement on that metric. A lot of things that seem true because they intuitively sound right with Euclidean distance break with the discrete metric.