Specifically, I am wondering if this proof requires that the second part is true for every compact subinterval, or whether it just needs to be true for an affine function that agrees with $f$ at endpoints.
To fully state what theorem I'm talking about :
Let $f$ be a real valued function defined on an interval $I$. then $f$ is convex IFF for every compact subinterval $j$ of $I$, and every affine function $L$, the supremum of $f+L$ is attained at an endpoint,
And to restate the question:
- does the result require this to be true for EVERY affine function $L$,?
- If it is not required, why is it included?