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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 perfect metric for finding weird counter examples

there 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.

Also "Counterexamples In Topology" - same but for topology.

I 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

yup, 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.

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