Yet there is no stationary point there. From left to right, the slope is sharply up on your side and sharply down on my side. Our derivative is discontinuous where we joined. That is the last exception to the rules. For the last case, the derivative must in fact pass through zero discontinuously .
Some folks call points like that cusps .
Perhaps we can go some place cozy and discuss higher derivatives at greater length. As promised, I will give you the true test for inflection points shortly. First, consider that we defined the second derivative of a function as the derivative of its derivative. In fact all the derivatives, fourth and upward, are zero. Note the notation for derivatives of order higher than second.