Definition of analytic curve double check.

[caffeinemachine]

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.

 

[jvicens]

Hello!

Your definitions seem to be correct. A real analytic function is indeed defined as a function that has a convergent Taylor series at each point in its domain, and a real analytic curve is a function whose components are all analytic. These definitions are commonly used in mathematics and are accepted as correct. However, it is always a good idea to double-check with a trusted source or consult a textbook to ensure accuracy. Keep up the good work!