From what I know, an inflection point is a point where the concavity changes.
However, I am wondering conceptually if there is a point $c\in Dom(f)$ where $f(c)$ is defined there but $f''(c)$ is undefined, but the concavity indeed changes at $c$, can $c$ be considered an inflection point or not?
Sure, how about $f(x) = \max \{x^2 - 1, 1 - x^2 \}$. It opens down from $-1$ to $1$, and opens up outside that interval. It's continuous everywhere, but neither the first nor the second derivative exist at $\pm 1$. If you want the first derivative to exist there as well, you have to work a little harder with how you pick the two functions so their tangents match up, but the same sort of piecewise thing will work.