6 Real line [09X]
We will indicate in the following with \(ℝ\) the real line, and with \(\overline{ℝ}=ℝ∪\{ +∞,-∞\} \) its extension. 1
We will use intervals (see definition in 67).
[2DJ]Given a set \(I⊂ℝ\) there are various ways of saying that a function \(f:I→ℝ\) is monotonic. Let’s first list the different types of monotonicity
Unfortunately in common use there are different and incompatible conventions used when naming the previous definitions. Here is a table, in which every convention is a column.
|
?? |
non-decreasing |
increasing |
weakly increasing |
|
?? |
increasing |
strictly increasing |
strictly increasing |
|
?? |
non-increasing |
decreasing |
weakly decreasing |
|
?? |
decreasing |
strictly decreasing |
strictly decreasing |
In this text, the convention in the last column is used.
(The first column is, in my opinion, problematic. It often leads to the use, unfortunately common, of phrases such as ”\(f\) is a non-decreasing function” or ”we take a function \(f\) not decreasing”; this can give rise to confusion: seems to say that \(f\) does not meet the requirement to be ”decreasing”, but it does not specify whether it is monotonic. People who follow the convention in the first column (in my opinion) should always say ”monotonic”).
- E173
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Show that any interval \(I\) in \(ℝ\) falls in one of the categories seen in 68. Hidden solution: [UNACCESSIBLE UUID ’09Z’]
- E173
[20V] Prerequisites:15.Let \(𝛼{\gt}0,𝛼∈ℝ\) be fixed. We know that, for every natural \(n≥ 1\), there exists an unique \(𝛽{\gt}0\) such that \(𝛽^ n=𝛼\), and \(𝛽\) is denoted by the notation \(\sqrt[n]{𝛼}\). (See e.g. Proposition 2.6.6 Chap. 2 Sec. 6 of the course notes [ 2 ] or Theorem 1.21 in [ 22 ] ). Given \(q∈ℚ\), we write \(q=n/m\) with \(n,m∈ℤ,m≥ 1\), we define
\[ 𝛼^{q}{\stackrel{.}{=}}\sqrt[m]{𝛼^ n}\quad . \]Show that this definition does not depend on the choice of representation \(q=n/m\); that
\[ 𝛼^{q}={\big({\sqrt[m]{𝛼}}\big)}^ n\quad ; \]that for \(p,q∈ℚ\)
\[ 𝛼^{q}𝛼^ p=𝛼^{p+q}\quad ,\quad (𝛼^ p)^ q=𝛼^{(pq)}\quad ; \]show that when \(𝛼 {\gt}1\) then \(p↦ 𝛼^ p\) is strictly monotonic increasing.
- E173
[20W] Prerequisites:2.Difficulty:*.Having fixed \(𝛼 {\gt}1\), we define, for \(x∈ℝ\),
\[ 𝛼 ^ x=\sup \{ 𝛼^ p : p∈ℚ, p≤ x\} \quad ; \]show that:
this is a good definition (i.e. that the set on the right is bounded above and not empty).
Iff \(x\) is rational then \(𝛼 ^ x\) (as above defined) coincides with the definition in the previous exercise 2.
show that \(x↦ 𝛼^ x\) is strictly increasing.
Show that
\[ 𝛼^{x}𝛼^ y=𝛼^{x+y}\quad ,\quad (𝛼^ x)^ y=𝛼^{(xy)}\quad . \]
See also the exercise 4.
- E173
[20X] Let \(a,b∈ℝ\) be such that
\[ ∀ L∈ℝ, L {\gt} b ⇒ L {\gt} a \quad . \]Prove that \(b≥ a\).
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[0B0] Fix \(I=\{ 1,\ldots n\} \). Let \(n\) distinct points \(y_ 1,\ldots y_ n∈ℝ\) be given; let \(𝜎:I→ I\) be a permutation for which triangle inequalities between successive points are equalities i.e.
\[ |y_{𝜎(i+2)}-y_{𝜎(i+1)}| + |y_{𝜎(i+1)}-y_{𝜎(i)} | =|y_{𝜎(i+2)}-y_{𝜎(i)} | \]for \(i=1,\ldots n-2\). Show that there are only two, we call them \(𝜎_ 1,𝜎_ 2\). Tip: Show that any such permutation necessarily puts the points ”in order”, i.e. you have
\[ ∀ i,y_{𝜎_ 1(i+1)}{\gt} y_{𝜎_ 1(i)}\quad ,\quad ∀ i,y_{𝜎_ 2(i+1)}{\lt} y_{𝜎_ 2(i)} \](up to deciding which is \(𝜎_ 1\) and which is \(𝜎_ 2\)).
Hidden solution: [UNACCESSIBLE UUID ’0B1’]