Let $f:[0,1]\to \mathbb{R}^n$ be a semi-analytic curve, then by [1], $f$ should be analytic everywhere with the exception of a finite number of points. I want to know whether $f$ is also piecewise analytic with a finite number of pieces. More specifically, 'piecewise analytic' means there exist $0=t_0<t_1<\cdots<t_m=1$ such that over any interval $[t_i,t_{i+1}]$, there exists an analytic function $g$ defined over an open superset of $[t_i,t_{i+1}]$ such that $f(x)=g(x)$ for all $x\in[t_i,t_{i+1}]$. The tricky part is at the boundaries. Ideally, I hope $f$ can also be represented as convergent power series in some interval $[t_i,t_i+\epsilon)$.
If the answer is no, is that possible to enforce some other properties to make $f$ piecewise analytic? Thanks so much!
some updates:
semi-analytic curve means the image of the curve is a semi-analytic set.
I found a Theorem in [2] on page 94, which says:
Theorem on the parametrization of a semi-analytic arc, stating that the germ of a semi-analytic arc at its end-point is the germ of the image of the interval $(0,\epsilon)$ under a non-constant analytic mapping of the interval $(−\epsilon, \epsilon)$, and conversely with a possibly smaller $\epsilon$.
In [2], the path is defined as a function $(0,1)\to\mathbb{R}^n$ so the theorem has $(0,\epsilon)$ in its statement.
It appears to me that the theorem above implies the following fact: the germ of $f|_{x\in[t_i,t_{i+1}]}$ at $t_i$ should be the same as the germ of the image of $[t_i,t_{i+1}]$ under some non-constant analytic function defined over an open superset of $[t_i,t_{i+1}]$. Then it seems to certify that $f$ is indeed 'piecewise analytic'. I am not an expert on this area so it is very likely I missed or misunderstood some part. I would really appreciate it if someone could help me either prove or find a counter-example.
[1] Gabrielov, Andrei M. "Projections of semi-analytic sets." Functional Analysis and its applications 2.4 (1968): 282-291.
[2] Łojasiewicz, Stanisław. "On semi-analytic and subanalytic geometry." Banach Center Publications 34.1 (1995): 89-104.