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Questions tagged [monotone-functions]

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. This tag may also include questions about applications or consequences of monotonicity, such as convergence, optimization, or inequalities.

In calculus, a function $f$ defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely increasing or decreasing. It is called monotonically increasing (also increasing or nondecreasing), if for all $x$ and $y$ such that $x \leq y$ one has $f(x) \leq f(y)$, so $f$ preserves the order. Likewise, a function is called monotonically decreasing (also decreasing or nonincreasing) if, whenever $x \leq y$, then $f(x)\geq f(y)$, so it reverses the order.

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Are Monotone functions Borel Measurable?

Could you guide me how to prove that any monotone function from $R\rightarrow R$ is Borel measurable? Since monotone functions are continuous away from countably many points, would that be helpful in proving the measurability?
user48405
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Can any analytic function be written as the difference of two monotonically increasing analytic functions?

This question is not a duplicate of this one or this one, since the solutions shown there contain jumps. Let $f(x)$ be an analytic function on $\mathbb{R}$. We can take its Taylor series and group the terms with a positive coefficient in one…
Kotlopou
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Proving monotonicity of a function $g$

Let $r\ge 1$. For $r-1\leq x \le r+1$ we define $f(x)=\arccos\left(\frac{x^2 + r^2 - 1}{2 r x}\right)$. Now let $g:[1,\infty)\to\mathbb{R}$ be given by $$g(r)=\frac{\int_{r-1}^{r+1}{r(f(x))^2\,dx}}{\int_{r-1}^{r+1}{ f(x)\,dx}}.$$ I want to show that…
Auslander
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Complete monotonicity of a sequence related to tetration

Let $\Delta$ denote the forward difference operator on a sequence: $$\Delta s_n = s_{n+1} - s_n,$$ and $\Delta^m$ denote the forward difference of the order $m$: $$\Delta^0 s_n = s_n, \quad \Delta^{m+1} s_n = \Delta\left(\Delta^m s_n\right).$$ We…
Vladimir Reshetnikov
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If $g:[0,1] \to \Bbb{R}$ such that $g(x)=g(y) \implies g'(x)=g'(y)$ for all $x,y \in (0,1)$, then $g$ is monotonic?

Is this true or false? Let $g:[0,1] \to \Bbb R$ be a function continuous on $[0,1]$ and differentiable on $(0,1)$, such that $g'$ is a function of $g$, i.e. for every $x, y \in (0,1)$, if $g(x) = g(y)$ then $g'(x) = g'(y)$. Then $g$ is…
Kenny Lau
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I'm looking for a function that is continuous and monotone increasing between a and b and maps to the entire real line.

After some trial-and-error I've got this: $$f(x;a,b)=\textrm{arctanh}\left( 2\frac{x-a}{b-a}-1 \right)$$ Illustrated here on WolframAlpha: arctanh((2*((x-a)/(b-a))-1)) where a=2 and b=12 It kind of does what I wanted, but I wonder if there are any…
André Christoffer Andersen
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If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing

PROBLEM. Let $f:[a,b]\to\mathbb R$ be a continuous function, such that $f'(x)\ge 0$, for all $x\in [a,b]\setminus A$, where $A\subset [a,b]$ is a countable set. Show that $f$ is increasing. Attention. In this problem, we DO NOT assume that $f$ is…
Yiorgos S. Smyrlis
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How to prove $x^3$ is strictly increasing

I am trying to use $f(x)=x^3$ as a counterexample to the following statement. If $f(x)$ is strictly increasing over $[a,b]$ then for any $x\in (a,b), f'(x)>0$. But how can I show that $f(x)=x^3$ is strictly increasing?
user133993
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Without any software and approximations prove that $\sec(52^{\circ})-\cos(52^{\circ})>1$

Without any software and approximations prove that $$\sec(52^{\circ})-\cos(52^{\circ})>1$$ We can use some known trig values like $18^{\circ}$,$54^{\circ}$,etc My try: I considered the function: $$f(x)=\sec(x)-\cos(x)-1,\: x\in \left (0,…
Ekaveera Gouribhatla
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Proof of Karlin-Rubin's theorem, detail about a real analysis fact.

Although the setting of this question is statistics, the question actually asks for a real analysis fact (monotone functions). Karlin-Rubin's theorem states conditions under which we can find a uniformly most powerful test (UMPT) for a statistical…
user39756
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When does this semimetric induced by a quasi-arithmetic mean fulfil the triangle inequality?

Definition. For a continuous, strictly monotone function $f \colon I \to J$, where $I, J \subset \mathbb R$ are intervals, we can define the $f$-mean of two numbers $p, q \in I$ as $$ M_f \colon I \to I, \qquad (p, q) \mapsto f^{-1}\left(…
ViktorStein
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Find all $x\in\mathbb{R}$ such that $\left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x$.

Find all $x\in\mathbb{R}$ such that: $$ \left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x\,. $$ Immediately we notice that $x=2$ satisfies the equation. Then we see that $LHS=a^x+b^x$, where $a<1$ and $b<2$, therefore $RHS$…
MartinYakuza
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Splitting a continuous monotonically-increasing function $f(x)$ as $h(x)+h(x+\epsilon) = f(x)$

Given a continuous monotonically-increasing function $f: [0,1]\to \mathbb{R}$ and a parameter $\epsilon>0$, does there exist a continuous monotonically-increasing function $h$ such that, for all $x\in[0,1]$: $$h(x)+h(x+\epsilon) = f(x)?$$ If…
Erel Segal-Halevi
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For every twice differentiable function $f : \bf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statements are TRUE?

For every twice differentiable function $f : \mathbf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statement(s) is (are) TRUE ? (A) There exist $r, s\in \bf R$, where $r < s$, such that $f$ is one-one on the open…
user3290550
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Proving monotonicity of this ratio of Hypergeometric functions

Let $n\in\Bbb N$, $\omega=0,1,\dots,n$, and $\nu,z>0$. We define $$ \tilde g_{n,\omega}(z,\nu):=\frac{z^{n-\omega}\partial_z^n z^\omega…
Aaron Hendrickson
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