Can a compact curve in $\mathbb{R}^n$ be characterised by the image of some analytic function?

Question

Given a curve in $\mathbb{R}^n$ that is non-intersecting and of finite length, more formally it is the image of an injective function $f(t):[0,1]\rightarrow\mathbb{R}^n$ with arc length $\int_0^1||f'(t)||dt<\infty$, can I give an alternative characterisation of the curve as the image of a real analytic function $g(t)$?