A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points where the second derivative is zero or doesn't exist. But if continuity is required in order for a point to be an inflection point, how can we consider points where the second derivative doesn't exist as inflection points?
Also, an inflection point is like a critical point except it isn't an extremum, correct? So why do we consider points where the second derivative doesn't exist as inflection points?