I want to show that for any curve $\gamma : [a,b]\to \mathbb{R}^2$ which is $C^1([a,b])$ is impossible that the range of $[\frac{\partial \gamma_1}{\partial t}, \frac{\partial \gamma_2}{\partial t}]$ is zero.
I honestly do not know where should I start, or even if it is true. Intuitively, it seems like if the derivative was $(0,0)$ then the function should stay constant in both directions, and I could not think of a geometric interpretation of that.
To give a bit of context, I wanted to show that the curve described has volume zero in $\mathbb{R}^2$. Since I know that the graphic of a function from $\mathbb{R}\to\mathbb{R}$ has measure zero, if I could show that every part of the curve is implicitly definable, then I would have won. But I am missing this step.