Can a point of inflection be undefined?

See the explanation section below.

A point of inflection is a point on the graph at which the concavity of the graph changes.

If a function is undefined at some value of ##x##, there can be no inflection point.

However, concavity can change as we pass, left to right across an ##x## values for which the function is undefined.

Example

##f(x) = 1/x## is concave down for ##x < 0## and concave up for ##x > 0##.

The concavity changes “at” ##x=0##.

But, since ##f(0)## is undefined, there is no inflection point for the graph of this function.

graph{1/x [-10.6, 11.9, -5.985, 5.265]}

Example 2

##f(x) = root3x## is concave up for ##x < 0## and concave down for ##x > 0##.

##f'(x) =1/(3root3x^2)## and ##f'(x) =(-2)/(9root3x^5)##

The second derivative is undefined at ##x=0##.

But, since ##f(0)## is defined, there is an inflection point for the graph of this function. Namely, ##(0,0)##

graph{x^(1/3) [-3.735, 5.034, -2.55, 1.835]}