Let $f:[0,\infty)\to \mathbb R ^2$ be a smooth (continuously differentiable forever) function.
Let $X$ be the image of $f$ (in the plane).
Say that $x\in X$ has unbounded curvature if there is a sequence of points $x_n\in X$ such that $x_n\to x$ and $\text{curv}(x_n)\to \infty$ as $n\to\infty$.
For instance, in the image below, $f(0)$ has unbounded curvature, according to my definition.
My question is, can $X$ have a dense set of points where the curvature is unbounded?