Earlier quoted context omitted.
Any continuous locus of points is a curve - that's what makes the Jordan Curve Theorem hard to prove :) On a more serious note, the graph displayed in the article is a timeseries of the difference between two points on the curve. The actual curve looks something like this: https://en.wikipedia.org/wiki/Yield_curve#/media/File:Yield_...
isn't that discontinuous?
Yield curves are bootstrapped from known bond yields at liquidly-traded tenors (1Y, 2Y, 5Y, 10Y, and short-dated). Those are the heavy black dots in the image. Outside of those tenors the 'true' rate is anyone's guess: you're basically interpolating. Whether that interpolation is continuous is up to you!