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.

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Monotone functions on $\mathbb{R}^n$

I am looking for references on the following topic. It is a known fact that, every measurable function $f:\mathbb{R} \to \mathbb{R}$ monotone can be seen as the sum $f = f_c + f_j$ where $f_c$ is a ...
Nestor Bravo's user avatar
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$(n + 1)$-cyclical monotonicity implies cyclical monotonicity in $\mathbb{R}^n$?

Say a function $f : \mathbb{R}^n \to \mathbb{R}^n$ is $N$-cyclically monotone if for any $x_1, \dots, x_{N + 1} \in \mathbb{R}^n$ with $x_1 = x_{N + 1},$ it holds that \begin{align} \sum_{i = 1}^{...
Paruru's user avatar
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Show that the jump $j_f(c)$ of increasing $f$ at $c$ is given by $\inf\{f(y)-f (x): x < c < y, x, y \in I\}$.

Let $I\subseteq \mathbb{R}$ be an interval and let $f: I \to \mathbb{R}$ be increasing on $I$. If $c$ is not an endpoint of $I$, show that the jump $j_f(c)$ of $f$ at $c$ is given by $\inf\{f(y)-f (x):...
user13's user avatar
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1 vote
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Monotonicity of $f$ and $f^{-1}$

Let $f$ be a strictly monotonic function on an interval $I$. Then, is $f^{-1}$ of the same monotonicity? I believe that the answer is YES. Here is my attempt: Since $f$ is monotonic in an interval, ...
jacie's user avatar
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6 votes
3 answers
133 views

If $f$ is continuous at $a$, is $f^{-1}$ continuous at $f(a)$?

Let $I\subseteq\mathbb{R}$ be an open interval, and $f:I\to\mathbb{R}$ an injective function. Let $a\in I$, and suppose that $f$ is continuous at $a$. Does it follow that $f^{-1}$ is continuous at $f(...
ashpool's user avatar
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Find necessary and sufficient conditions for ordinal monotonicity.

First of all let's we remember the following result. Theorem Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ ...
Antonio Maria Di Mauro's user avatar
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Direction changes in product of functions

Suppose $f(x) = g(x) h(x)$ and the number of direction changes in both $g$ and $h$ are known. Is there a simple way to prove that the number of direction changes in $f$ is bounded by the number of ...
Tommy Tang's user avatar
2 votes
1 answer
68 views

Is monotonicity hold of a continuous vector function across union of sets? [closed]

A vector function $F$ is monotonic in $S_i$ if $$ (F(x)-F(y))^T (x-y)\geq 0, \forall x,y\in S_i. $$ However, I am unsure if this property holds for continuous functions when considering the union of ...
Yuzhen Feng's user avatar
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Find all $x \in \mathbb{R}$ such that $f: \mathbb{N} \rightarrow \mathbb{R}$, where $f(n) = \{2^n \cdot x\}$ is monotone on $\mathbb{N}$.

Find all $x \in \mathbb{R}$ such that $f: \mathbb{N} \rightarrow \mathbb{R}$, where $f(n) = \{2^n \cdot x\}$ (where $\{x\}$ denotes the fractional part of $x$), is increasing/decreasing on $\mathbb{N}$...
math.enthusiast9's user avatar
2 votes
1 answer
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A question about the monotonicity of $y_1 = x^e$ and $y_2 = x^{\pi}$

As we know from 1st year Calculus, for function $f(x) = x^n, x \in \mathbb{R}, n \in \mathbb{N}$, if $n$ is odd, the $f(x)$ is increasing; if $n$ is even $f(x)$ is increasing when $x \geq 0$ and ...
ZYX's user avatar
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Clamp endpoint derivatives of cubic polynomail such that the polynomial becomes monotonic on interval

I have a polynomial $P(x) = Ax^3 + Bx^2 + Cx + D$, with $P(0) = 1$ and $P(1) = 0$. This means that $D=1$ and $A + B + C + D = 0$. Suppose that $P'(0) = d_0 \leq 0$ and $P'(1) = d_1 \leq 0$. Depending ...
user877329's user avatar
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Classifing functions with the property that the preimage of path-connected sets is path-connected

Given two path-connected topological spaces $\mathcal{X},\mathcal{Y}$, consider a function $f: \mathcal{X} \to \mathcal{Y}$ with the property, that the preimage $f^{-1}(V)$ is pathconnected for all ...
allmendum's user avatar
1 vote
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How to prove a function is strictly increasing?

Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is ...
cclinoom's user avatar
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Proving the sequence of harmonic function uniformly converges on compact sets.

Fix an open, bounded $U \subset \mathbb{R}^n$ and consider a sequence $C^2(U) \cap C(\bar{U})$ of harmonic functions. (i) Suppose that $u_n$ converges uniformly to a function $u.$ Show that u is ...
Lilili123's user avatar
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Monotonicity of the probability of a sum of independent random variables being below a threshold

Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. For $A>0$, is the function $$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A) $$ monotonically ...
ilanshom's user avatar
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1 answer
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When is a sequence that is bounded below by an unbounded and monotone increasing also monotone increasing? [closed]

Let $(a_n)$ be an unbounded and monotone increasing sequence of positive integers and let $(b_n)$ be another sequence of positive integers such that for all $n \in \mathbb{N}$, $a_n \leq b_n$. ...
trillianhaze's user avatar
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2 answers
88 views

$x^n/(1+x^n) $is monotonic

$x^n/(1+x^n)$ is monotonic , $x \in R^+ $ , $n \in N$ tried solving it by differentiating but not getting feasible result, also is there any other way of computation through which this can be solved ...
Svidi Runs's user avatar
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Number of monotonically increasing functions such that $f(i)\le i$.

Problem: Consider $n \in \mathbb{N}^+$, set $A = \mathbb{N}^+ _{\leq n}$. Find the number of monotonically increasing functions $f: A → A $ such that $f(i) \leq i$. I tried using the multinomial ...
Trulaug's user avatar
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Monotonicity of an expectation of monotonous function with unknown closed-form.

Suppose that we are given an arbitrary set $\mathcal X \subseteq \mathbb R^d$, a probability density function of multivariate Gaussian distribution with mean $\mu$ and covariance matrix $\sigma^2I$, f$...
Interception's user avatar
2 votes
2 answers
87 views

Can a discontinuous function be increasing or decreasing

How would we decide increasing or decreasing function if the function is not differentiable? Can a discontinuous function be increasing or decreasing? And can it be monotonic?
S K's user avatar
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First derivative test - how to prove it from the "monotonicity theorem"?

Here are two related theorems. I have trouble proving the second one. Monotonicity theorem: Let $f$ be a real valued function defined on $\mathbb R$. The function is continuous at a point $a$ and $f$ ...
niobium's user avatar
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Is there a monotone function that is not differentiable everywhere? [duplicate]

One of the most strange things that I learned in Real Analysis is a function that is continuous on $\mathbb{R}$ but that is not differentiable everywhere. I wonder how to prove that no monotone ...
pie's user avatar
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Monotonic relationship with plateaus and vertical portions?

For a monotonic relationship between $x$ and $y$, an increase $x$ always leads to an increase in $y$ or always leads to a decrease in $y$. On a graph, the slope is always positive or always negative. ...
user2153235's user avatar
3 votes
1 answer
310 views

Some doubtful implication for mathematical analysis.

Let, $f(x),g(x),f_1(x),g_1(x)$ are positive real valued bounded and continuous functions on domain of non-negative reals and also having range between $0$ and $1$. And, also, $f_1(x),g_1(x)$ are ...
A learner's user avatar
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Is there a purely algebraic proof that the sequence $a_N:=\left(\frac{N!}{N^N}\right)^\frac{1}{N} $ is monotone decreasing?

I saw this question and I quickly noticed that the first inequality (after using $e^x$ on both sides) $\frac{N^N}{e^N}< N!$ can be written as $\frac{N!}{N^N}> e^{-N}$ or $\left(\frac{N!}{N^N}\...
pie's user avatar
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Concave continous function with special property implies that is strictly increasing?

I have the following problem. Let $\rho(s)\geq 0$ for all $s\geq 0$ be a continuous concave function, satisfying $\rho(s)\geq\rho_1(s)$ for all $s>0$, where $\rho_1$ is a strictly increasing and ...
Sergio Garcia Castro's user avatar
3 votes
1 answer
69 views

Show that the recursive sequence $x_{k+1} = |x_k - \frac{x_k}{1-2M^2x_k^2}|$ is monotone

I'm doing some exercises for an upcoming exam, and as part of a larger problem, I want to show that the given recursive sequence: $$x_{k+1} = \left|x_k - \frac{x_k}{1-2M^2x_k^2}\right|$$ is monotonly ...
maibrl's user avatar
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1 vote
0 answers
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Question regarding monotone relations

I read on a paper that a relation $\mathcal{R} \subset \mathbb{R}^n \times \mathbb{R}^n$ is monotone if \begin{equation} (x_1 - x_2)^\top (y_1 - y_2) \geq 0 \end{equation} holds for any pair $(x_1, ...
Trb2's user avatar
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2 votes
1 answer
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Proving that this sigmoid sequence converges

The problem Given $h_0=0.6$ I want to prove that the following iterative sequence converges. I also want to find the value it converges to. $$h_{t+1}= \sigma(3h_t-1)=\frac{1}{1+e^{-(3h_t-1)}}$$ My ...
John Katsantas's user avatar
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0 answers
10 views

Monotonic family of monotonic functions

I am imagining a family of monotonic functions, which can be represented as a parametric function $f(\cdot; \tau): \mathbb R \to \mathbb R$ parameterized by $\tau \in \mathbb R$ such that (1) when $\...
Vezen BU's user avatar
  • 2,016
4 votes
1 answer
114 views

Equivalent condition for the following monotone-like condition for multivariate function

Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$ be a continuously twice differentiable function. I am considering the following condition (1) on $f$: For any $x,x'\in\mathbb R^n$, If it is ...
Andeanlll's user avatar
1 vote
1 answer
110 views

Monotonicity of Convex Conjugate Divided by Quadratic

I came across the following (unproved) convex analysis fact in a paper. Fact: Suppose $\varphi:[0,\infty)\to\mathbb{R}$ is a convex function. Define its convex conjugate $\varphi^*(y)=\sup_{x\ge 0}[...
Kittayo's user avatar
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8 votes
0 answers
179 views

Dirichlet's test with unimodal coefficients

Briefly: If we modify the hypotheses of Dirichlet's test to require a unimodal sequence of coefficients, not necessarily a monotonic sequence, then do we still get the same quantitative bound on $\sum ...
Chris Culter's user avatar
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Monotonicity of a function involving Normal cdf

I am trying to prove the monotonicity of the following function. $$f(x) = x + \frac{\Phi'\left(x\right)}{\Phi\left(x\right)}$$ for $x\in \mathbb{R}$ where $\Phi\left(x\right)$ is normal cdf and $\Phi'...
mike's user avatar
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4 votes
4 answers
284 views

How to prove that $f(x)=\frac{ \cot(\frac{\pi}{x+1}) }{ \cot(\frac{\pi}{x}) }\cdot\frac{x}{x+1}$ is strictly deceasing for $x>2$?

I read an article that claimed that among two regular polygons with equal perimeters, the one with more sides has a larger area, in other words $\frac{ \cot(\frac{\pi}{n+1}) }{ \cot(\frac{\pi}{n}) }\...
pie's user avatar
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3 votes
1 answer
199 views

Monotonicity of a multivariate function

Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$. Assume $f(x,y)$ is twise continuously differentiable and the derivative with respect to both arguments (i.e., $x$ and $y$) are monotone in ...
Andeanlll's user avatar
1 vote
1 answer
129 views

Positive definiteness of a cross-derivative equivalent condition

Let $f:\mathbb R^n\times\mathbb R^n\rightarrow \mathbb R$ be a twice continuously differentiable function. The cross-derivative of $f$ is denoted by $D_{xy}f$. I want to check whether the following ...
Andeanlll's user avatar
2 votes
1 answer
146 views

Name of this monotone-like property of a function

There is a function $f: \mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$. $f$ satisfies the following property for any $x\neq x'$ and $y\neq y'$: $$f(x,y')>f(x,y)~\textrm{implies}~f(x',y')\geq f(...
Andeanlll's user avatar
1 vote
1 answer
86 views

Is the set of non-decreasing functions from $[0,1]$ to a compact subset of $\mathbb{R}$ compact and first countable?

Let $Y$ be a compact subset of $\mathbb{R}$, and $X\subseteq Y^{[0, 1]}$ be the set of non-decreasing functions from $[0,1]$ to $Y$, where $Y^{[0,1]}$ has product topology. Is $X$ compact and first-...
qscty's user avatar
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1 vote
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Name for functions over metric VS that preserve scalar product inequalities?

Let $V_1$ and $V_2$ be two metric vector spaces over $\mathbb{R}$. Let $f: V_1 \rightarrow V_2$ with the following property: $\forall \vec{x},\vec{y},\vec{z} \in V_1$ such that $\vec{x} \cdot \vec{y} \...
Ywen's user avatar
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Why doesn't Euclid prove uniqueness of a chord from the fact that length is strictly decreasing?

In III.7 Euclid proves that $FA > FB > FC > FG > FE$ using the Triangle Inequality and Hinge Theorem. He then goes on to prove using triangles that if $P$ is a point on the side of the ...
SRobertJames's user avatar
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0 votes
1 answer
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In what way knowing that $f$ is increasing helps me here?

I have to show that: given $f: A\to \mathbb{R}$ and $g: B\to A$ two real functions, then if $f$ is concave and increasing in $A$ and $g$ is concave in $B$, the composition $f\circ g$ is concave in $B$....
Heidegger's user avatar
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When to include the boundary points in the convexity analysis?

I was wondering about this: suppose I have a function $f: D \to \mathbb{R}$; suppose $(a, b) \subset D$ ($D$ can either be bounded or unbounded), and say $f$ is convex in $(a, b)$. What is the ...
Heidegger's user avatar
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1 vote
1 answer
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A claim regarding some summations of monotonic functions in fraction

I am trying to prove this claim but it seems the math somehow does not work out... Let $f_1,f_2,g_1$ and $g_2$ be real-valued, strictly positive, continuously differentiable and strictly decreasing ...
Paul H.Y. Cheung's user avatar
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28 views

What approach to take to parameterise a C2, monotonic, piecewise polynomial?

I have a set of points (x, y) and need to monotonically interpolate between them. The resulting polynomial needs to be C2 continuous. Could you point me in the ...
Hiperfly's user avatar
1 vote
0 answers
19 views

Proving the monotonicity of a recursively defined sequence of functions

I am trying to prove that the following sequence of function $$p_{n+1}(x) = p_n(x) + \frac{x^2-p_n^2(x)}{2}, \; (n \geq 0)$$ with $p_0=0$ is monotonically increasing on the interval $[-1,1]$. My ...
geom_groom's user avatar
6 votes
3 answers
298 views

prove that $f’(x)e^{\lambda x}$ is increasing if and only if $f’(x)+\lambda f(x)$ is increasing. Where $f\in C^1(0,\infty)$.

prove that $f’(x)e^{\lambda x}$ is increasing if and only if $f’(x)+\lambda f(x)$ is increasing. Where $f\in C^1(0,\infty)$, and $\lambda$ is a real number. I have tried to prove it by taking $0<...
xp D's user avatar
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1 vote
1 answer
51 views

Problem regarding increasing/decreasing function

Let $ f:(0,\frac{\pi}{2})\rightarrow\mathbb{R}$ be given by $f(x)=(\sin x)^\pi-\pi \sin x+\pi$. Then which of the following are true?a. f is an increasing function.b. f is a decreasing function.c. $...
Shreya Jaganathan's user avatar
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31 views

Change the monotonicity of a function but preserve its convexity

Consider the function $f : \mathbb R^2 \to \mathbb R$ defined as $$ f(x) = \exp(x) + \exp(y) $$ $f$ is convex in $(x,y)$ and it is strictly monotone increasing in $(x,y)$ (in the sense that, if $x_1 &...
mhdadk's user avatar
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Sufficient condition for the characteristic function of a discrete distribution to be decreasing

Let $X$ be a discrete random variable defined on the lattice such that $P(X \in \mathbb{Z}/h) = 1$, where $\mathbb{Z}/h$ represents the set of integers scaled by a factor of $h$. I wonder if there ...
XiaoHei's user avatar
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