Do you agree, at least, that there’s no need to require a tangent line at a POI?

]]>I don’t really have a solution that I find totally satisfactory.

]]>What requirement would you choose, Will?

]]>I suspect that if you asked someone 200 years ago what a relative maximum was, you’d get something like this: A relative maximum is a point where a function stops increasing and starts decreasing. But in our attempts to define a relative maximum precisely (and in our relatively liberal attitude to what counts as a function), we are forced to accept relative maxima such as when and when and the fact that constant functions have relative maxima everywhere.

Similarly, perhaps we have an intuitive idea of what a point of inflection is: it’s a point where a function stops speeding up and starts slowing down or stops slowing down and starts speeding up.

If your primary goal is to characterize these points for what they say about what is happening to the rate at which the function is changing, then you could see why you might insist that, for example, doesn’t have a point of inflection at .

If, on the other hand, you’re simply interested in keeping track of where the concavity changes, then I don’t see why you’d even insist on continuity. After all, as the above example shows, we don’t insist on continuity to classify a point as a relative maximum.

]]>But I’m not sure you’ve highlighted the “larger problem” when it comes to POI, unless you demand that there be a tangent line at a POI. Do you? I don’t, so I discard immediately any discussion of tangent lines.

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