Convex functions

Whenever we talk about convex functions, an inequality suddenly arise that presents an algebraic definition for a geometric content. I mean we have an image of convexity and from that imagination to this definition there is something to explain. The only thing I never have understood is that “what does it come from?” “Why we define a convex function as something that satisfies in such inequality?” The below image is just an example of a mathematical book (Rudin, complex and real analysis) but you can find it in every mathematical book, and in all cases a very same sentence is repeating even with no extra comma!? Can you tell me why we define a convex function as something that satisfies in that inequality?

 

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