Can point of inflection occur at a point where second derivative doesnot exist?

I refered these questions Definition of Point of Inflection An inflection point where the second derivative doesn't exist? What is inflection point? I couldnt understand the necessary conditions for point of inflection.My specific doubt is Can point of inflection occur at a point where the second derivative of a function is discontinuous? In the image attached, the curves of the actual function, its second derivative and third derivative are given(from left to right correspondingly) Can the point x=b be an inflection point for the function.

• You have answered your own question. math.stackexchange.com/questions/402459/… Yes a point where the second derivative is discontinuous at that point can be an inflection point. – Matthew Liu Feb 1 at 7:34
• @Matthew Liu Ok.Thanks for answering.Does this mean the point x=b in the diagram can be a point of inflection for f(x) ? – Mohan Feb 1 at 7:43
• I think it "point of inflection" is only defined as change in sign of the second derivative and that this change of sign does not need to be continuous and go through 0. But it was ages ago I even saw this expression. – mathreadler Feb 1 at 8:06
• @mathreadler A point of inflection is a geometric property. It's a point where the tangent line crosses the graph. – B. Goddard Feb 1 at 11:39
• @B.Goddard Hmm if you say so. I think last time I encountered the name was in high school... Definitely did not occur particularly often at university for me. – mathreadler Feb 1 at 13:12

Take $$y=\sqrt[3]{x}$$. Then $$(0,0)$$ is an inflection point, but the second derivative is discontinuous there.