# Real analyticity of continuous function via restriction to analytic curves

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.

• 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 $f(x,y) = y^2-x^3$. May 1 '21 at 12:22
• Thanks a lot Jason! I wonder is this statement at least correct in case when $X$ is smooth? May 1 '21 at 13:56
• You might be interested in the following article of J'anos Koll'ar and Krzysztof Nowak: arxiv.org/abs/1301.5048 May 1 '21 at 19:55

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.