I'm studying analytic functions and wondering if there is any version of Open Mapping Theorem for real analytic functions.

In particular, let $f:\mathbb{R}^m\to\mathbb{R}^n$ be a real analytic function. Let $D\subset\mathbb{R}^m$ be an open subset. Is it true that $f(D)$ is also an open subset of $\mathbb{R}^n$? I have seen this kind of result for complex analytic functions but not sure if it also holds for real functions.

If this does not hold for analytic functions then which types of real functions can have this property?

I wd appreciate if someone can provide some references on this topic.


  • 1
    $\begingroup$ What about $f(x) = x^2$ on $(-1,1)$? $\endgroup$ – copper.hat Jan 4 '17 at 16:03
  • 2
    $\begingroup$ No, for all sorts of reasons: 1. things like $\cos x$ 2. The function $(x,y) \to (x,0)$ 3. Real analytic functions can have maximums $\endgroup$ – zhw. Jan 4 '17 at 16:05
  • $\begingroup$ are you aware of any real functions from R^m to R^n that might have this property? $\endgroup$ – jayki Jan 4 '17 at 17:14

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