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Consider the function $f:\mathcal{R}^n\longrightarrow\mathcal{R}^m$, where $\mathcal{R}$ denotes the set of real numbers and $m$ and $n$ are any two different positive natural numbers. Suppose that the Jacobian matrix of $f$ exists for each $x\in\mathcal{R}^n$.

  1. Which conditions on $f$ ensure that its jacobian has a right inverse on $\mathcal{R}^n$?
  2. Conversely, if the Jacobian of $f$ has a right inverse on $\mathcal{R}^n$, then what can we say about $f$?
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We assume that $f\in C^1$. $Jac(f)$ has a right inverse in $X_0$ IFF $Jac(f)$ is surjective in $X_0$; then $Jac(f)$ is surjective in $U$, a neighborhood of $X_0$ and $f_{|U}$ is an open application. The converse is true.

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