Real analyticity of a $C^{\infty }$-germ at the origin

The work is devoted to real analyticity in the ball $B(0,1)$ of a function $f$ that is infinitely smooth at $0\in {\mathbb R}^{n}$ and whose restrictions to lines $l\ni 0$ are real analytic in the interval $l\cap B(0,1)\approx (-1,1).$

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