Simple question right?
This website, along with the Calc book we’re teaching from, define it this way:
A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.
There’s no debate about functions like , which has an unambiguous inflection point at .
In fact, I think we’re all in agreement that:
- There has to be a change in concavity. That is, we require that for we have and for we have , or vice versa.*
- The original function has to be continuous at . That is, does not have a point of inflection at even though there’s a concavity change because isn’t even defined here. If we then piecewise-define so that it carries the same values except at for which we define , we still don’t consider this a point of inflection because of the lack of continuity.
But the part of the definition that requires to have a tangent line is problematic, in my opinion. I know why they say it this way, of course. They want to capture functions that have a concavity change across a vertical tangent line, such as . Here we have a concavity change (concave up to concave down) across and there is a tangent line () but is undefined.
So It’s clear that this definition is built to include vertical tangents. It’s also obvious that the definition is built in such a way as to exclude cusps and corners. Why? What’s wrong with a cusp or corner being a point of inflection? I would claim that the piecewise-defined function shown above has a point of inflection at even though no tangent line exists here. [Edit: Originally I had used as my example, but as my dad so astutely pointed out, that does have a tangent line at zero. Doh! Thanks dad!]
I prefer the definition:
A point where the graph of a function is continuous and where the concavity changes is a point of inflection.
That is, I would only require the two conditions listed at the beginning of this post. What do you think?
Once you’re done thinking about that, consider this strange example that has no point of inflection even though there’s a concavity change. As my colleague Matt suggests, could we consider this a region of inflection? Now we’re just being silly, right?
* When we say that a function is concave up or down on a certain interval, we actually mean or for the whole interval except at finitely many locations. If there are point discontinuities, we still consider the interval to have the same concavity.
** This source, interestingly, seems to require differentiability at the point. I think most of us would agree this is too strong a requirement, right?