Is it possible to construct a closed, simple curve in $\mathbb{R}^2$ that has a segment with zero curvature and is differentiable everywhere?

Question

The question title says it all. Presume I am interested in creating a closed, simple curve in $\mathbb{R}^2$ that contains a segment with zero curvature (that is, part of it is a straight line). This segment must have finite length. I am now interested in whether or not it is possible to create a closed, simple curve with such a segment that is also differentiable. Since a curve made up of only line segments (a polygon) would not be differentiable at the vertices, I expect that such a curve would be made of nonlinear curves stitched in somehow so that the derivatives are equal at the stitching points, perhaps? I'm not sure how to proceed to create such a curve, or what it might look like.

Answer 1

Take the left half of the circle radius $1$ centered at the origin, and the right half of the circle radius $1$ centered at $(1,0)$ and add to these the line segments from $(0,-1)$ to $(1,-1)$ and $(0,1)$ to $(1,1)$, i.e., join the two half circles together.