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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?

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  • $\begingroup$ If a function is concave and convex everywhere, you could argue that the concavity doesn't actually "change" in any point. $\endgroup$ – StackTD May 13 '16 at 7:37
  • $\begingroup$ @StackTD: How might we precisely define the word "change" to capture your idea? $\endgroup$ – Kenny LJ May 13 '16 at 7:38
  • $\begingroup$ What I mean is: you could change the definition(s), like you say, but perhaps an interpretation of the current one is sufficient to not have to call every point an inflection point. To me, it makes sense to say that the concavity changes in a point if the concavity is different on both sides of that point. In this case, the concavity is identical in every point - nothing changes. $\endgroup$ – StackTD May 13 '16 at 7:41
  • $\begingroup$ Wikipedia defines points of inflection as isolated extremum of the first derivative. Roughly speaking, this means $f''$ changes sign about the POI. This would preclude linear functions having POIs everywhere. $\endgroup$ – MathematicsStudent1122 May 13 '16 at 8:20
  • $\begingroup$ @MathematicsStudent1122: But what if the function is not differentiable? How then might we write down the correct definition of a point of inflection? $\endgroup$ – Kenny LJ May 13 '16 at 8:41
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A linear function is both convex and concave at the same and at every point, so according to the definition you have written down every point is an inflection point.

Another definition I came across requires a change in concavity, which, in fact, for a linear function does not occur (it has the same concavity behaviour everywhere). So you should be careful how you write down the defnition.

In general there is some freedom in mathematical definitions, and not all mathematical definitions are used consistently throughout the community (another example are different sign conventions for certain quantities, like the Laplacian). What is important is that you make precise what you are talking about, so if you are using a definition for which you think you found a problem like in this case you should make sure to point out which formulation you are using.

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