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In my lecture notes about Inverse Function Theorem it said

There is no need to have an Inverse Function Theorem for functions $F : \mathbb R^n → \mathbb R^m$. If we only require $F$ and $F^{-1}$ to be continuous (not necessarily differentiable), then we can still conclude m = n."

Does it mean that for $F: \mathbb R^n → \mathbb R^m$ which has an inverse function $F^{-1}:\mathbb R^m → \mathbb R^n$, says $F$ is differentiable at $a\in \mathbb R^n$ and $F^{-1}$ is differentiable at $b=F(a)\in \mathbb R^m$,then $m$ must be equal to $n$? Why is it true?

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The Inverse Function Theorem requires a continuously differentiable function.

Your assumption is true. The Invariance of Domain Theorem says that $F$ cannot be bijective and continuous if $m\neq n$.

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