For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function. Or an commutative monoid without inverses. Or a compact topological space that is countable.

In cases where the object has been proven or conjectured to not exist, it could bring up the theorem or conjecture that states so. For example, if I search for differentiable non-continuous functions, it would bring up this theorem perhaps.

Does such a thing (or something close to it) exist?

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    $\begingroup$ I don't know of anything quite like that, but I may as well mention The On-Line Encyclopedia of Integer Sequences, which is extremely useful from time to time. $\endgroup$ – Eric Tressler Jun 24 '16 at 23:25
  • $\begingroup$ You may be looking for books of counterexamples. For example "counterexamples in analysis" by Gelbaum and Olmsted may be of interest. $\endgroup$ – Sean English Jun 24 '16 at 23:29
  • $\begingroup$ For topology there is topology.jdabbs.com, where you can search spaces in general topology. It's based on the book "counterexamples in topology". In general it makes sense to search by subfield of mathematics. $\endgroup$ – Henno Brandsma Jun 25 '16 at 7:00
  • $\begingroup$ I think this question is too broad, but if you will restrict it there may be some answers in some categories. For example, a very incomplete list is at github.com/alex-konovalov/gnu/wiki/Mathematical-databases $\endgroup$ – Alexander Konovalov Jun 26 '16 at 6:33
  • $\begingroup$ There are thousands of such databases scattered around the world, but unfortunately mathematicians carry them inside their heads, so you have to query them through an online interfa...oh. $\endgroup$ – Andrew D. Hwang Jun 26 '16 at 15:00

This database exists and is called the web. Googling

  1. Continuous non-differentiable function leads to Weierstrass function:

In mathematics, the Weierstrass function is an example of a pathological real-valued function on the real line. The function has the property of being continuous everywhere but differentiable nowhere.

  1. Smooth but non-analytical function leads to Non-analytic smooth function, where you can find several detailed examples.

  2. Countable compact topological space leads to Countable compact spaces as ordinals, a beautiful answer on this site.

Theorem 4. Every countable compact Hausdorff space is homeomorphic to some well-ordered set with the order topology.

  1. I have to confess that Commutative monoid without inverses does not give a direct link, probably because $0$ is always its own inverse. But $(\mathbb{N}, +, 0)$ is a standard example of a commutative monoid in which no nonzero element has an inverse.

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