Point of inflexion. Given a function, $$f(x)=\frac{1}{x-2}$$ Do we say that a point of inflexion exists at $x=2$? Because on either side, the concavity changes.

Your function is not defined for $x=2$ so there is not a point on his graph with $x=2$. So there is not an inflection point of the graph of the function.

So, the concavity can change also if there is not an inflection point!