Curvature and Torsion of a Curve

Question

I am currently working on a problem and think I know the answer but need verification. The question reads:

Let there be a curve with non-zero curvature and zero torsion. Show this curve is planar. If the curve is allowed zero curvature at one point, does this above statement still hold?

I have shown that the curve is planar with non-zero curvature and zero torsion. But when the curve has zero curvature $\textit{and}$ zero torsion, isn't the curve a straight line there? And if so, doesn't this straight line remain in the original plane normal to the constant $\boldsymbol b$?

Answer 1

Let $$f(t)=\begin{cases}(t,t^3,0)&t\leq 0\\(t,0,t^3)&t\geq 0\end{cases}$$ Here $f$ has zero torsion but is not planar. It has zero curvature only at $t=0$.