Let $\gamma: [a,b]\to\mathbb{R}^d$ be a smooth (i.e. $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve with arc-length defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)).$$

Assume that $\gamma_j(t),\gamma_j^\prime(t)\neq 0$ for $1\le j\le d$.

Define a partition $P=\{x_0,x_1,\dots,x_n\}$ on $[a,b]$ by: $$ x_m=\begin{cases} a & m=0\\ b & m=n \\ \max_{t\in[x_{m-1},b]}\{\forall s\in [a,t],\gamma(s)\neq \gamma(t)\} & \text{otherwise} \end{cases}. $$ Roughly speaking, this partition divides $\gamma(t)$ to injective graphs in respect to last coordinate.

My question is "how large can P be?". To put it in other words, Can we find a curve $\gamma$ with $|P| = \aleph_0?\aleph?$ (where $|P|$ denotes cardinality of the set).