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Math Help - Function 1-1 thorughout, if its 1-1 on an open set.

  1. #1
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    Function 1-1 thorughout, if its 1-1 on an open set.

    Let f:\mathbb{R}^{n} \to \mathbb{R}^{n} be a continuous function which is 1-1 on any open set in \mathbb{R}^{n}. Prove that f is 1-1 throughout the domain.
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    Let f:\mathbb{R}^{n} \to \mathbb{R}^{n} be a continuous function which is 1-1 on any open set in \mathbb{R}^{n}. Prove that f is 1-1 throughout the domain.

    Solution 1 (The wiseguy solution): \mathbb{R}^n is open in itself and thus, according to the given data,  f is 1-1 there.

    Solution 2: \mathbb{R}^n-\{0\} is open so f is 1-1 there. Now, if f(0)=f(x_1) , for some x_1\in\mathbb{R}^n ,then take \mathbb{R}^n-\{x_2\}\,,\,\,x_2\neq 0\,,\,x_1 ...

    By the way , the continuity of f isn't required.

    Tonio
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