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A corollary to Hurwitz' theorem from complex analysis states the following:

Corollary (Hurwitz). Suppose $f_N:\Omega\to\mathbb{C}$ is holomorphic with $f_N\neq 0$ on $\Omega$, and $f_N\to f$ uniformly on compact subsets of $\Omega$. Then either $f\equiv 0$ or $f(z)\neq 0$ for all $z\in\Omega$.

I was wondering if there was a similar or related theorem for a real-valued sequence of functions $g_N:\Omega\to\mathbb{R}$, where $\Omega\subset\mathbb{C}$ ?

After some thought I could see that the corollary cannot be used for real-valued functions due to the holomorphic condition, but wondered if there were any analogous theorems from real analysis?

pshmath0
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    If $f$ is real analytic on a connected open set :-). – copper.hat Jul 12 '17 at 16:26
  • @copper.hat thanks for your comment. Are you saying that if $f(x,y)$ is real analytic on a connected open set and $f_n(x,y)\neq 0$ for all $n$ and $f_n\to f$ uniformly then $f\equiv 0$ or $f\neq 0$ ? Do you have a link or reference where I could learn more... – pshmath0 Jul 12 '17 at 16:32
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    Well, basically it can be extended to a holomorphic function on some set containing the interval in question so it would follow from the above result (see https://math.stackexchange.com/a/1034715/27978, for example). – copper.hat Jul 12 '17 at 16:41
  • Ah I see, yes I had entertained that thought, thanks. – pshmath0 Jul 12 '17 at 16:52

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