According to Wikipedia and also ProofWiki, a real-valued function $f$ is concave if, for any $x_1$ and $x_2$ in the interval and for any $\alpha\in[0,1]$, $$f(\alpha x_1 + (1-\alpha)x_2) \geq \alpha x_1 + (1-\alpha)x_2.$$ (The definition of convex is analogous, just change $\geq$ to $\leq$.)
An inflection point is then a point where $f$ changes from being concave to convex (or vice versa).
Here's where I have a doubt: Given any linear function, the function is everywhere both concave and convex. So wouldn't every point of any such function be an inflection point? But my impression is that we don't want to say that every point of a linear function is an inflection point.
So to resolve this, should the correct definition instead be that "an inflection point is where $f$ changes from being strictly concave to strictly convex (or vice versa)"? (Or perhaps something else?)
Or there's nothing wrong with Wikipedia/ProofWiki and I'm missing something?