Curve and sweeping line intersection: continuity

Question

Let $\gamma$ be a simple, piecewise differentiable, closed plane curve.
Assume that the slope of the tangent to $\gamma$, where defined, is never $m$.
Show that the function $ x \mapsto \sup \{ ||\mathrm{u_2} - \mathrm{u_1}||, \ \mathrm{u_1}, \mathrm{u_2} \in \gamma \cap \mathcal{l} (m,x)\}$ is continuous over $\mathbb{R}$.
Here, $\mathcal{l} (m,c)$ is the line in the plane having slope $m$ and intercept $c$, while $||\circ||$ is the Euclidean distance.

I think I can prove this elementarily, by picking a parametrization for $\gamma$ and working with delta/epsilons and standard analysis. However, it would be nice if there were any other less sweaty ways of proving (or disproving!) this, or even merely alternative ways. I would be bewildered if there were no more elegant framework in which this result is dealt with immediately.

Answer 1

My own claim is false.

As a counterexample, consider the outer contour of this figure as $\gamma$:

Take the y=x line and sweep.

For the ultimate application I need, I think I can still use a related result, but I don't know how to state it rigorously.