A curve with Lebesgue measure non zero

Question

In this Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact image. For example I could take the curve(with dense image in $[-1,1]^2$) $$\gamma(t)=(\cos(t),\cos(\sqrt2t)),$$ for $t\in\mathbb{R}$.