= Local diffeomorphism
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A local diffeomorphism is a <differentiable map> that restricts near every point to a <diffeomorphism> onto an <open subset> of its target. The local <inverse functions> need not combine into a single global <inverse function>: different parts of the <domain of a function> can map onto the same output region.
For <continuously differentiable> maps between <open subsets> of $\mathbb R^n$, being a <local diffeomorphism> is equivalent to having an invertible <derivative> at every point. The <inverse function theorem> proves one direction. For the other, the <chain rule> for each local <inverse function> gives two inverse <linear maps>. The periodic angle example under the <inverse function theorem> has this property everywhere, even though it has infinitely many inputs for every point of the punctured plane. The <noninjective map with constant Jacobian determinant> gives another useful distinction: even a constant nonzero <Jacobian determinant> need not give global <injectivity>.
Every <local diffeomorphism> is an <open map>. To see this, take an <open subset> of the source and intersect it with the neighbourhoods where the map is a <diffeomorphism>. Each local image is <open>, and their <union> is the full image. Thus nearby outputs remain available locally, despite the possible global ambiguity.
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