# First- and second-order discontinuities

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?

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It’s a point at which the derivative is discontinuous—if you want to refer to it ordinally, I’d call it “second-order”, because the convention for ordinal indices is to start counting from one. (As much as I’d say that in general zero-indexing is more appropriate…)

Giving them short, kind-of-memorable-but-not-really ordinal names is well and fine, but if you want short names then yeah, just go with “jumps” and “kinks”. You can couple that with full descriptions of what exactly these mean: a “jump” is a discontinuity (of the type that is sometimes referred to as a “jump discontinuity”, as opposed to an asymptote or a hole), and a “kink” is a (jump) discontinuity in the derivative.

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Thanks @zzq! It now occurs to me that one could also just be slightly vague and call the kinks “higher-order discontinuities”. But I think you’re right that a first-order discontinuity is just a discontinuity, i.e., where \lim\limits_{x\to c} f(x) \ne f(c).

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