I find conflicting reports on this which leads me to believe there may be conflicting opinions or varying explanations among textbooks.
Can a point of inflection be identified where the function has a vertical asymptote just because the concavity changes? For example does y=1/x have a point of inflection at x=0? My belief is that a point of inflection cannot exist at a point where the function is not defined or even not differentiable.

Question

I find conflicting reports on this which leads me to believe there may be conflicting opinions or varying explanations among textbooks.
Can a point of inflection be identified where the function has a vertical asymptote just because the concavity changes?  For example does y=1/x have a point of inflection at x=0?  My belief is that a point of inflection cannot exist at a point where the function is not defined or even not differentiable.

anonymous · Answer

i agree with you

anonymous · Answer

It depends on what domain you're working. On the Real numbers, you are correct. It is not defined. On the extended real numbers (which allow infinity and negative infinity), it is defined as an inflection point. That's probably why your textbooks have different answers.

directrix · Answer

A friend e-mailed the following:

I would argue that y=x^(1/3) has a point of inflection at the origin, even though the function is not differentiable there. While a function that changes concavity via a "corner" does not have a point of inflection at said point. "My" defn of an inflection point is a "smooth change in concavity". This allows for vertical tangents, but not cusps and certainly not points of discontinuity.
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My take on his thinking is that he is creating his own definitions.

anonymous · Answer

By its very definition, a point of inflection must exist at a point where a function's second derivative is either zero or undefined. So yes, points of inflection may exist where a function does not. Read http://www.sosmath.com/calculus/diff/der15/der15.html for more details.

anonymous · Answer

I also would have said that if the concavity change results from discontinuity/asymptote then that is not a point of inflection.

directrix · Answer

Another mate's thoughts on this:

If you use the English (as in "England," not US English), the word
is "inflexion," which connotes "not flexed." Using this logic, there would
have to be a (unique) tangent at a point of inflection, although the
tangent could be vertical. 

I prefer a simpler definition, a point of inflection is a point
(on the graph, not just an x-value) at which the concavity changes sign.