Should function be continuous at point of inflection

Question

I have some very fundamental doubts regarding point of inflection.

Should function be continuous at point of inflection.

Should derivative exist at point of inflection.

No book tells about these fundamental things but wax eloquent for differentiable functions on this topic.

Answer 1

In order for a point of inflection to exist, the first derivative must reach an extremum at that point . But for the derivative to reach an extremum, the first derivative must exist in the first place, and in order for the first derivative to exist somewhere, the function must be continuous there as well.

Answer 2

Yes, the function must be differentiable (and therefore continuous) at the point of inflection. The point $x_0\in D_f$ is an inflection point of the function $f$ if $f'$ has a local extreme at $x_0$.