I understand if a morphism of schemes $f : X \to Y$ is locally of finite type by Hartshorne's definition, then $f$ is locally of finite type by the Stacks project's definition.
I'm working on formalizing locally contractible spaces in Mathlib (the mathematics library for the Lean theorem prover), and I've encountered conflicting terminology in the literature regarding local
A topological space is locally connected if every point admits a neighbourhood basis consisting of open connected sets. To the definition given by Lee (Introduction to topological manifolds - page $92$) which sums up definitions $1$ and $2$ more "compactly" as follows;
The definition is given in the third paragraph on the Wikipedia article (also see the following paragraph for differences in naming). Note that local conformal flatness is a property of Riemannian manifold, so you need to specify a Riemannian metric. Amazingly, every 2-dimensional Riemannian manifold is locally conformally flat - this is the theorem you are referring to.
The statement is indeed true, and this sheaf is usually denoted by $j_ {!}\mathcal {F}$ where $j:A \hookrightarrow X$ is the inclusion of the locally closed subset $A \subset X$.
Let $f\colon \mathbb R^n\to \mathbb R$ be a function that is locally point Lipschitz at $0$, not locally Lipschitz at $0$, and locally Lipschitz everywhere else, as you have described.
If we can show that any affine scheme (locally of finite type etc.) contains a smooth open dense subset, then the union of those open dense subsets for the $V_i$ will give an open dense smooth subset of $X$.
Understanding proof that If $X$ is locally of finite type and ...