Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$?

Remark 1. The answer to the above question is well-known to be negative by basic results in topological dimension theory if the condition that $\Gamma_f$ is closed is strengthened to the continuity of $f$.

Remark 2. By elementary topology, a function $f$ on a compact Hausdorff space is continuous iff $\Gamma_f$ is closed. This fact can be used to show that the answer to the above question is also negative if in the statement of the question, $\Bbb{R}$ is replaced by $[0,1]$, i.e., $f$ is stipulated to be a function from $[0, 1]^m$ to $[0,1]^n$.

Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$?

Remark 1. The answer to the above question is well-known to be negative by basic results in topological dimension theory if the condition that $\Gamma_f$ is closed is strengthened to the continuity of $f$.

Remark 2. By elementary topology, a function $f$ from a compact Hausdorff space $X$ to a compact Hausdorff space $Y$ is continuous iff $\Gamma_f$ is closed in $X \times Y$. This fact can be used to show that the answer to the above question is also negative if in the statement of the question, $\Bbb{R}$ is replaced by $[0,1]$, i.e., $f$ is stipulated to be a function from $[0, 1]^m$ to $[0,1]^n$.