I'm having a hard time visualizing the gimbal lock problem. Suppose $p=(a,b,c)$ is a point where the euler angle $f:\mathbb R^3\to SO_3$ has zero jacobian determinant. Then to say that $p$ is where gimbal lock occurs can be mathematically formulated as saying that $f$ has no bijection from a small area around $p$ to a small ball around $f(p)$.
So there are rotations very close to $f(p)$ which cannot be represented as $f(p+\epsilon)$ for small $\epsilon$. I am having a hard time visualizing this. Does someone have a way to see that such rotations cannot be represented as $f(p+\epsilon)$?