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Suppose $X\subset \mathbb R^n$ is an irreducible real analytic sub-variety (i.e. the set of solutions of a system $f_1=\ldots=f_k=0$ with $f_i$ analytic)

Let $x\in X$ be a point and let $F: X\to \mathbb R^1$ be a continuous function defined on $X$ in a neighbourhood of $x$. I want to understand whether $F$ is real analytic on $X$. The question is whether the following would be sufficient to know.

Property. Suppose that for any real analytic map $\varphi: (-1,1)\to X$ sending $0$ to $x$ the composition $F\circ \varphi$ is analytic on $(-1,1)$.

Question. Does it follow from the property that $F$ is real analytic in a neighbourhood of $x$?

I am interested both in positive statements in this direction (possibly strengthening the condition of the Property) and in counterexamples.

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    $\begingroup$ That is not true. Consider the continuous inverse $F$ of the following real analytic homeomorphism $G$ from $\mathbb{R}$ to its image $X$ inside $\mathbb{R}^2$, namely $G(t) = (t^2,t^3)$ where $X$ is the zero set of $f(x,y) = y^2-x^3$. $\endgroup$ May 1, 2021 at 12:22
  • $\begingroup$ Thanks a lot Jason! I wonder is this statement at least correct in case when $X$ is smooth? $\endgroup$
    – aglearner
    May 1, 2021 at 13:56
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    $\begingroup$ You might be interested in the following article of J'anos Koll'ar and Krzysztof Nowak: arxiv.org/abs/1301.5048 $\endgroup$ May 1, 2021 at 19:55

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No. Take $X = \mathbb R^2$ and $F(x,y) = \frac{x^3}{x^2+y^2}$. Then $F$ is real-analytic everywhere but $(0,0)$. Moreover, at $(0,0)$, any curve passing through $(0,0)$ must have coordinates two analytic functions $x,y$ vanishing to orders $a,b$, in which case $x^2+y^2$ vanishes to order $2\min(a,b)$ and $x^3$ vanishes to order $3a > 2 \min(a,b)$ so the ratio is a well-defined analytic function.

But $F$ is not analytic at $(0,0)$.

A similar trick can be used to construct worse functions, like the irrational $F(x,y) = \frac{ x^5}{ \sqrt{ (x^2+y^2) (x^2 + 2y^2)}} $ and, by summing terms of this form, functions that fail to be analytic at many points.

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