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Definition of analytic curve double check.

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Hello MHB.
Can someone please check if these definitions are correct.

Definition.
Let $U$ be a subset of the real numbers. A function $f:U\to\mathbb R$ is said to be a real analytic function if $f$ has a Taylor series about each point $x\in U$ that converges to the function $f$ in an open neighborhood of $x$.

Definition.
Let $U$ be a subset of real numbers and $f:U\to\mathbb R^n$ be a function. Write $f(x)=(f_1(x),\ldots,f_n(x))$. Then $f$ is said to be a real analytic curve if each $f_i$ is are analytic.