Applying the inverse function theorem, we can quite easily show that the function $$ f \colon \mathbb{R}^2 \rightarrow \mathbb{R}^2\quad (x,y)\mapsto \begin{pmatrix}x \exp(y) \\ y \exp(x) \end{pmatrix} $$ is locally invertible away from the points where the Jacobian matrix $Df$ is not invertible. But what about the other points, i.e. those where $xy=1$, lying on the hyperbola $y = \frac{1}{x}$? Is $f$ invertible on a small neighborhood? Or might $f$ even be injective (which I failed to show either)?
Local inverse/injectivity of a function involving products of $\exp$
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real-analysis
exponential-function