15.3 Real case

Let \(I⊂ ℝ\), then \(I\) is convex if and only it is an interval (see 1). In the following we will consider \(f:I→ℝ\) where \(I=(a,b)\) is an open interval.

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[18F] Show that \(f(x)\) is convex if and only if the map \(R(x,y)=\frac{f(x)-f(y)}{x-y}\) is monotonically weakly increasing in \(x\). 1 Moreover, \(f\) is strictly convex if and only if \(R\) is strictly increasing. Hidden solution: [UNACCESSIBLE UUID ’18G’]

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[18H]Show that for a convex function \(f:(a,b)→ℝ\) there are only three possibilities:

  • \(f\) is strictly increasing

  • \(f\) is strictly decreasing

  • There are two values \(l_-≤ l_+\) such that \(f\) is strictly increasing in \([l_+,b)\), \(f\) is strictly decreasing in \((a,l_-]\), and the interval \([l_-,l_+]\) are all minimum points of \(f\);

If also \(f\) is strictly convex then there is at most only one minimum point.

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[18J]Let \(f : (a, b)→ ℝ\) be convex. Show that, for every closed interval \(I ⊂ (a, b)\), there exists a constant \(C\) such that \(f |_ I\) is Lipschitz with constant \(C\). Provide an example of a continuous and convex function defined on a closed interval that is not Lipschitz.

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[18K]Prove that a continuous function \(f : (a, b) → ℝ\) is convex if and only if, for every \(u, v ∈ (a, b)\),

\[ f\left(\frac{u+v} 2\right) ≤ \frac{f (u) + f (v)} 2\quad . \]

Convexity and derivatives

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[18M] Prerequisites:1.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.

Hidden solution: [UNACCESSIBLE UUID ’18N’]

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[18P] Prerequisites:1.If \(f: (a, b)→ ℝ\) is differentiable, then \(f\) is convex if and only if \(f'\) is weakly increasing. Hidden solution: [UNACCESSIBLE UUID ’18Q’]

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[18R] Prerequisites:1,2.If \(f: (a, b)→ ℝ\) is differentiable, then \(f\) is strictly convex, if and only if \(f'\) is strictly increasing. Hidden solution: [UNACCESSIBLE UUID ’18S’]

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[18T] Prerequisites:1, 2.Suppose that \(f: (a, b)→ ℝ\) is twice differentiable. \(f\) is convex if and only if \(f''≥ 0\) at every point. Hidden solution: [UNACCESSIBLE UUID ’18V’]

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[18W] Prerequisites:4.

Let \(J⊂ ℝ \) be an open nonempty interval, and \(f:J→ℝ\) be a twice differentiable and convex function. Show that the following facts are equivalent:

  1. \(f\) is strictly convex,

  2. the set \(\{ x∈ J:f''(x)=0\} \) has an empty interior,

  3. \(f'\) is monotonic strictly increasing.

Hidden solution: [UNACCESSIBLE UUID ’18X’]

See also the exercise 4 for the relationship between integral and convexity.

Convex functions with extended values

We consider convex functions that can also take on value \(+∞.\) Let \(I\) be an interval.

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[18Y]Let \(f:I→ℝ∪\{ ∞\} \) be convex, show that \(J=\{ x∈ I:f(x){\lt}∞\} \) is an interval.

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[18Z]Note:another vice versa of 2.

Given \(I⊆ ℝ\) interval and \(f:I→ℝ∪\{ ∞\} \) convex and lower semicontinuous, there exist sequences \(a_ n,b_ n∈ℝ\) such that \(f(x)=\sup _ n (a_ n+b_ n x)\).

Hidden solution: [UNACCESSIBLE UUID ’190’]

  1. Note that \(R(x,y)\) is symmetrical.