From the blog:
Yellow Brick Roads are defined as piecewise-linear functions and we need a name for the points where the slope changes. The mathematical term “inflection point” is commonly abused to mean any sudden change in slope of a graph. The actual mathematical term we want is “second-order discontinuity” or “critical point”. As is also common in math, we’re using the word “kink” since its primary definition — “a sharp twist or curve in something that is otherwise straight” — captures intuitively what we mean. (Yes, we feel slightly weird about it.) If the kink is a transition from positive to negative slope we call it a peak and for negative to positive, a valley. Most kinks are just sudden changes in slope, neither a peak nor valley.
This came up when working on fixing do-more ratcheting and now I can’t decide if that’s quite right what we said about discontinuities in the blog post! Maybe a jump discontinuity is a zeroth-order discontinuity and a kink is a first-order discontinuity? Google is being difficult about this.
Can we get a real mathematician on the case?