EDB — 18M

[18M] Prerequisites:[18F]↺↻.Let \(f: (a, b)→ ℝ\) be convex.

1. Show that, at every point, right derivative \(d^+(x)\) and left derivative \(d^-(x)\) exist (In particular \(f\) is continuous).

2. Show that \(d^-(x)≤ d^+(x)\),

3. while, for \(x{\lt} y\), \(d^+(x)≤ R(x,y) ≤ d^-(y)\).

4. hence \(d^+(x)\) and \(d^-(x)\) are monotonic weakly increasing.

5. Show that \(d^+(x)\) is right continuous, while \(d^-(x)\) is left continuous.

6. Also show that \(\lim _{s→ x-}d^+(s)=d^-(x)\), while \(\lim _{s→ x+}d^-(s)=d^+(x)\). In particular \(d^+\) is continuous in \(x\), if and only if \(d^-\) is continuous in \(x\), if and only if \(d^-(x)= d^+(x)\).

   So \(d^+,d^-\) are, so to speak, the same monotonic function, with the exception that, at any point of discontinuity, \(d^+\) assumes the value of the right limit while \(d^-\) the value of the left limit.

7. Use the above to show that \(f\) is differentiable in \(x\) if and only if \(d^+\) is continuous in \(x\), if and only if \(d^-\) is continuous in \(x\).

8. Eventually, prove that \(f\) is differentiable, except in a countable number of points.