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Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?

Notice: when $f$ is $C^1$, we can use geometry techniques. When $f$ satisfies $$\limsup \frac{|f(y)-f(x)|}{|y-x|} < \infty$$ for any $x \in \mathbb{R}^n$, we can consider the measure of $f(\mathbb{R}^n)$. Unfortunately, the above conditions may not be met.

I can’t understand why this question is considered an undergraduate level question. In fact, facing this problem, many techniques in real analysis, differential geometry and topology have failed. I have not come up with a good solution.

Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?

Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?

Notice: when $f$ is $C^1$, we can use geometry techniques. When $f$ satisfies $$\limsup \frac{|f(y)-f(x)|}{|y-x|} < \infty$$ for any $x \in \mathbb{R}^n$, we can consider the measure of $f(\mathbb{R}^n)$. Unfortunately, the above conditions are not necessarily met.

I can’t understand why this question is considered an undergraduate level question. In fact, facing this problem, many techniques in real analysis, differential geometry and topology have failed. I have not come up with a good solution.

Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?

Notice: when $f$ is $C^1$, we can use geometry techniques. When $f$ satisfies $$\limsup \frac{|f(y)-f(x)|}{|y-x|} < \infty$$ for any $x \in \mathbb{R}^n$, we can consider the measure of $f(\mathbb{R}^n)$. Unfortunately, the above conditions may not be met.

I can’t understand why this question is considered an undergraduate level question. In fact, facing this problem, many techniques in real analysis, differential geometry and topology have failed. I have not come up with a good solution.

Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?

Notice: when $f$ is $C^1$, we can use geometry techniques. When $f$ satisfies $$\limsup \frac{|f(y)-f(x)|}{|y-x|} < \infty$$ for any $x \in \mathbb{R}^n$, we can consider the measure of $f(\mathbb{R}^n)$. Unfortunately, the above conditions are not necessarily met.

I can’t understand why this question is considered an undergraduate level question.In fact, facing this problem, many techniques in real analysis, differential geometry and topology have failed. I have not come up with a good solution.

Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?

Notice: when $f$ is $C^1$, we can use geometry techniques. When $f$ satisfies $$\limsup \frac{|f(y)-f(x)|}{|y-x|} < \infty$$ for any $x \in \mathbb{R}^n$, we can consider the measure of $f(\mathbb{R}^n)$. Unfortunately, the above conditions are not necessarily met.

I can’t understand why this question is considered an undergraduate level question. In fact, facing this problem, many techniques in real analysis, differential geometry and topology have failed. I have not come up with a good solution.