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Assume that I am given two varieties $X,Y$ and a map $\phi:X\to Y$ such that $\phi\mid_{X^{smth}}:X^{smth}\to Y^{smth}$ is an isomorphism and also $\phi\mid_{X^{sgl}}:X^{sgl}\to Y^{sgl}$ is an isomorphism, where $X^{smth}$ denotes the smooth locus and $X^{sgl}$ denotes the singular locus. An example of such a map would be a piecewise defined map, defined on the smooth locus as some isomorphism and on the singular locus as another isomorphism.

I expect that such a map is just not very well-defined so it should not be a morphism. Unfortunately, I was unable to come up with a counterexample. Can someone help out?

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Let $A = k[t^3,t^4] \subset k[t]$ and $B = k[t^2,t^3] \subset k[t]$ and set $X = Spec(A)$, $Y = Spec(B)$. The embedding $A \to B$ induces a morphism $\psi \colon Y \to X$ which induces isomorphisms of smooth and singular loci of $X$ and $Y$, but itself is not an isomorphism. Consequently, the "inverse" of $\psi$ is a map with the required properties which is not a morphism.

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