# Factoids #2

2020-12-26, post № 238

**mathematics**, #bijection, #calculus, #Lipschitz, #naturals

## VII) Cardinality coercion:

Claim. There exist both together with .

Proof. Iota. Define via and observe any sequence’s reconstructibility by dyadic expansion.Pi. Define via and observe any real’s constructibility by dyadic expansion.

Thus, in is shown.

## VIII) Codomain crumpling:

Let .

Claim. There exists .

Proof. Define via and observe any -sequence’s reconstructibility by counting -segmented section’s lengths as well as any -sequence’s constructibility by defining -run lengths.

## IX) A non-isomorphic Lipschitzian bijection

Claim. A bijective Lipschitzian map need not be an isomorphism in with Lipschitzian arrows, i. e. its inverse may degrade in regularity.

Proof. LetBy and the arctangent’s strict isotonicity together with ’s conventional parametrization, ’s bijectivity is apparent. Since is furthermore smooth, is globally Lipschitzian for all . As the local Lipschitz constant approaches zero for , one can deduce ’s global Lipschitz continuity.

Now, looking at a neighborhood of one finds for any given a radius with its ball satisfying , proving ’s non-continuity and thereby its violation of the Lipschitzian property.

Now, looking at a neighborhood of one finds for any given a radius with its ball satisfying , proving ’s non-continuity and thereby its violation of the Lipschitzian property.