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.

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Prove that $h(u)=\frac{2}{u^2}(u-\log(1+u))$ is decreasing on $(-1,\infty)$

Prove that $h(u)=\frac{2}{u^2}(u-\log(1+u))$ is decreasing on $(-1,\infty)- \{0\}$ This function appears on the proof of Stirling's formula in Principles of Mathematical Analysis, by Walter Rudin. ...
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Do I have enough information to prove this function is increasing?

Let $\alpha>0$ be a parameter and consider some function $f_\alpha(x,y)$ where $x\in(0,2\alpha)$ and $y\in(0,1)$. I have deduced the following properties for $f_\alpha$: $f_\alpha(x,y)>0$. $f_\...
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1answer
26 views

Median of $f(x)$ = $f($median of $x)$ if $f$ is monotonic?

Suppose $X \in \mathbb{R}$ is a continuous random variable, $f: \mathbb{R} \Rightarrow \mathbb{R}$ is monotonic. Let $Y = f(X)$, can we claim that $median(Y)=f(median(X))$, or median of a monotonic ...
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A doubt about differentiability and increasing function

Suppose $f:(a,b)\rightarrow (a,b)$ is differentiable on $(a,b)$ and for an $x_{0}$ such that $a<x_{0}<b$ , $f'(x_{0}) >0 $ then is $f$ increasing in some neighborhood of $x_{0}$? I have seen ...
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15 views

Monotony of a digamma(psi) function

Are $ f(x)=x\Psi(x)$ and $f_2(x)=x^2\Psi(x)$ monotonic i.e. increase when $x\in(0,+\infty)$?
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An less commonly seen sufficient condition for convexity

I was reading a proof in which the author claimed that, for a function $f(x)$ on $(0,1)$ which is continuous and strictly increasing, the following condition implies that convexity holds. Let $A > ...
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1answer
27 views

Given a continuous monotone function $f$, is $f^{-1}(x)$ a connected set?

Given a continuous monotone function $f: \mathbb{R} \rightarrow \mathbb{R}$, is it true that for any point $x \in \mathbb{R}$, $f^{-1}(x)$ must also be connected? The monotonicity is defined as a non ...
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1answer
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convex and unbounded implies increasing?

Suppose $f(x)$ is a positive continuous function on $(-\infty,\infty)$, symmetric about $0$. Let $f(x)$ is convex and $\lim_{x \rightarrow \infty} f(x) = \infty$. Can we say that $f(x)$ has to be ...
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Ways to fit a monotonic function to the data?

Asked at SO but it seems no one have an idea, so I'm coming again. Suppose I have 3 data points ($(0, 0), (0 < t_2 < 1, 0 < y_2 < 1), (1, 1))$ and I want to use a monotonic function to ...
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Monotone function and inverse function

Show that the function $y= f(x)= 1+ x^{2} + \arctan x^{2}$ is strictly monotone in $[0,+\infty ]$. If $f^{-1}$ is the reverse function of $f$, calculate the limit $$\lim_{y\rightarrow 1+}\frac{f^{-1}(...
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Prove monotonicity of function

Let $K>0$, $\gamma>0$, and $\tau_{xL}>\tau_{xH}>0$. We define: $$\tau^\star_{\epsilon H}=\frac{(\gamma^2+K\tau_{xL})-\sqrt{(\gamma^2+K\tau_{xH})(\gamma^2+K\tau_{xL})}}{\tau_{xL}-\tau_{xH}}$...
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2answers
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A simpler way to prove that $-x-\frac{1}{x+1}\leq-\frac{3}{2}$ for $x\geq1$, instead of computing derivative?

I have this function $$f(x)=-x-\frac{1}{x+1}$$ I want to prove that $f(x)\leq -\frac{3}{2}$ for $x\geq1$. We can easily prove this by calculating the first derivative which is negative, thus, the ...
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1answer
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Prove that if $f$ is a continuous map which maps every open set to an open set then $f$ is monotonic.

My attempt:If the function is not monotonous then we can consider points $a < b$ and $b <c$ such that $f(a)<f(b)$ and $f(b)>f(c)$.My intuition behind this was to consider a subset $[a,c]$ ...
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How many 6 digit positive integers have their digits in weakly decreasing order?

How do I solve this combinatorically? I'm not quite sure how to approach this since each digit is dependent on the digit before it. For example, if the first digit is a 9, then there are 10 possible ...
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Prove that $\exists$ $f:(a,b)\to [a,b]$ such that $f$ is Bijective. [duplicate]

Let $a,b \in \Bbb{R}$ such that $a<b$ Then Prove that $\:\exists\:$ a continuous function $f:(a,b)\to [a,b]$ such that $f$ is Bijective. I am not sure how to prove formally, But i tested the claim ...
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1answer
36 views

Prove that there exists $\gamma \in (a,b)$ such that $f \circ f(\gamma)=\gamma$

If $a,b,c \in R$ such that $a<b<c$ and $f:R \to R$ is a continuous function satisfying $f(a)=b,f(b)=c,f(c)=a$, then there exists $\gamma \in (a,b)$ such that $(f \circ f)(\gamma)=\gamma$ My try: ...
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Is a recursively given series monotone?

The series is given as $a_{n+1}=\frac{8a_{n}-3}{2a_{n}+1}$ and $a_{1}=1$ I have no idea how to prove if this series is monotone or not. I have tried rearranging the equation to form $a_{n+1}>a_{n}$ ...
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1answer
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Existence of neighborhood monotonic increasing

I was thinking about the following statement: Supose $f\in C^1$ a function s.t. $f(b)>f(a) \forall b\in (a,a+\epsilon)$. So, there's exist a neighborhood $(a,a+\delta)$ s.t. $f$ is monotonic ...
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Is it true that if $\forall n \in {N}$ $b_n \gt a_n \land b_{n+1} - b_n \lt 0 \implies a_{n+1} - a_n \lt 0$?

Let's say that I have some sequences: $b_n, a_n$. Note that: $\forall n \in N$ $b_n \gt a_n$. Assume I'm able to say easily that $b_n$ does not increase, so can I conclude that $a_n$ does not increase ...
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Are such functions increasing?

Let $g:\mathbb{R}\to \mathbb{R}$ be a function satisfying the intermediate value property (Darboux property) and consider $h:\mathbb{R}^2\to \mathbb{R}$ with $\min\{x,y\}<h(x,y)<\max\{x,y\}$ (...
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Prove that the given function is monotonic

Can someone please prove the following statement? A bijective function $f:[a,b] \to [c,d] $ is always a monotone one (i.e. it is either monotonically increasing or monotonically decreasing.) Assume ...
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3answers
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A continuous and derivable decreasing function s.t. the derivative is $0$ for $t>t_0$?

The question may be silly, but I am stucked with this thought... I ask for a nonconstant continuous and derivable nondecreasing function s.t. the derivative approaches $0$ for $t\to\infty$, but more ...
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1answer
20 views

Monotony for a not derivable function by parts

Let $f(x)=\begin{cases} \frac{(e^{x^2}-5x^2-2)}{x^2}, \quad x\neq 0 \\ 0 , \quad x=0 \end{cases}$ This is not a derivable function in fact it is not continous since $\lim_{...
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37 views

What are examples of discrete functions that are monotonically increasing/decreasing?

What are examples of discrete functions that are monotonically increasing/decreasing? Can monotonically increasing/decreasing be defined over discrete functions?
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1answer
40 views

If f has no local extrema, then it is monotonic

Let $f : \mathbb R \to \mathbb R$ continuous without local extremas. Show that $f$ is monotonic. Here is my attempt, proving the contraposition : If $f$ is not monotonic then $\exists a < b < c \...
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23 views

Prove that if a function is monotonic in [a,b] or (a,b), the limit of every x from this segment or interval exists.

Okay, so this is actually given in my textbook, but I can't understand it. Let's suppose that $f(x)$ is a monotonically increasing(not decreasing) function and $c$ is a random point such that $a<c&...
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Cyclically Monotone Functions of Vectors

I am using the following definition of cyclically monotonicity: on a set $\Gamma$, for all $(x_1, y_1)$ and $(x_2, y_2)$ if $x_1 \leq x_2$ then $y_1 \leq y_2$. My question is about how this definition ...
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1answer
20 views

Properties, bounds and limits about difference of two inverse standard normal CDF variables and extreme value distribution

I'm interested in the variable: $$\sigma_n=\Phi^{-1}\left(1-{1\over n}e^{-1}\right)-\Phi^{-1}\left(1-{1\over n}\right),$$ where $\Phi(\cdot)$ is the CDF of standard normal distribution. I want to ...
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1answer
45 views

What makes a bound different than a limit? (sequences) [closed]

I just started learning about sequences, and recently got to the boundedness part. One of my homework problems asked me to find a limit, and the answer that I got was $6$. I did not know how to find ...
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20 views

Does uniform convergence preserve strict monotonicity?

Motivated by this post, I would like to understand the following. If a sequence of strictly increasing functions converges uniformly to some function, then can we claim that this function is also ...
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3answers
115 views

Prove $\lim\limits_{x\to0} xf(x) = 0$?

I am confused about the following proof on a textbook I'm reading. Suppose $f:(0,b] \to \Bbb R$ is continuous, positive, and integrable on $(0,b]$. Suppose further that as $x \to 0$ from the right, $f(...
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1answer
28 views

Finding the intersecting points of two strictly monotonic increasing functions

I need to prove this: Let $f$ be a strictly monotonic continuous real-valued function defined on $[a,b]$ such that $f(a)<a$ and $f(b)>b$, then $\exists$ exactly one c $\in$ $(a,b)$ such that $f(...
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2answers
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On the slope of a strictly increasing function

If $f$ is a strictly increasing function, shouldn't $f'$ be always positive and never zero? Apparently there's this situation where the derivative can be $0$ if it's only at discrete points and not an ...
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35 views

Monotonicity of an expression involving derivatives

Let $p$ be a positive integer and $y$ be a function with class $C^p$ on some neighbourhood of $0$ in the real line. Define for any real positive number $\kappa$ $$ H_p(\kappa) = \sum_{i=1}^p \left\{[\...
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Is it true that $f(x)=k ,k\in \mathbb{R}$ is a decreasing and increasing and constant function in the same time over it's domain of definition?

It is known that $f(x)=k,k\in \mathbb{R}$ is a constant function ,one of my friend argued me that he has an example of functions which it is increasing and decreasing and constant in the same time ...
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Is there a systematic way to find a left adjoint of a given monotone function

What's the process by which one can find the left adjoint for a given monotone function? For example, excercise 1.95 in Dr. Spivak and Dr. Fong's book "An Invitation to Applied Category Theory&...
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67 views

Formal definition of a monotonically increasing function

My professor is working with the definition that a function is monotonically increasing when $x_1<x_2. \implies f(x_1)<f(x_2)$. Is this definition correct? And most importantly, if we already ...
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Discrete Completely Monotonic Function

A function $f:[1,\infty)\to \mathbf{R}$ is called Completely Monotonic Function if $(-1)^k f^k (x) \leq 0$ for every $k=1,2,\ldots$, so the signs of the derivatives of $f$ change signs. For example: $\...
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1answer
19 views

Can a bounded function be monotonically increasing for all x>0?

Can we have a function such that it is always increasing for all x>0 and is also bounded above by some real number? How do we prove or disprove this statement?(The above statement is not a problem ...
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Left continuity from some condition involving infimum

Let $g:[0,1]\rightarrow [0,1]$ be an increasing function, $E$ a (sequentially) complete space and such that there is some $f:E\rightarrow [0,1]$ continuous and $h:E\rightarrow [0,1]$ with $$g(x) = \...
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How to show that the sum of a function is strictly increasing?

I have that $\{r_1,r_2,r_3,...\}$ is an enumeration of $Q$ and let $ f_n(x)= \begin{cases} \frac{1}{n^2}&\text{if}\, x > r_n\\ 0 &\text{if}\, x \le r_n \end{cases} \quad$ and $f(x)=\sum_{n=...
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Compute inf and sup of a set A

Compute the infimum and supremum of $A=\{f(x)=\frac{2x+1}{x+2}: x>-2\}$. I try to do these following passages: since $f$ is derivable I compute $f'(x)=\frac{3}{x+2}>0$ and so from Monotone ...
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5answers
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Monotone Sequence $\frac{n}{n+1}$

I have to proof this sequence : $(x_n) := \frac{n}{n+1}$ for $n\in \Bbb N^+$ is a monotone sequence. I saw some examples of people using the induction method, so I tried it, but I got stuck when I was ...
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0answers
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Using induction with monotonicity to show that $F^n(\overrightarrow{\emptyset}) \sqsubseteq \overrightarrow{RD}$

This question is related to this recent question. The difference is that this question relates to a different proof of a different claim by the authors. Specifically, this question presents an ...
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Using induction with monotonicity to show that $F^n(\overrightarrow{\emptyset}) \sqsubseteq F^{n + 1}(\overrightarrow{\emptyset})$

This question is related to this recent question. The difference is that this question relates to a different proof of a different claim by the authors. Specifically, this question presents an ...
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1answer
32 views

Cardinality of order preserving functions from totally ordered set with dense subset

I know that in the category of continuous functions, if $X$ and $Y$ are Hausdorff topological spaces and $D\subseteq X$ is a dense subset of $X$, then the cardinality of the continuous functions from $...
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32 views

How do we know that $F^{n + 1}(\overrightarrow{\emptyset}) = F(F^n(\overrightarrow{\emptyset}))$?

I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Chapter 1.3 Data Flow Analysis says the following: The least solution. The ...
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0answers
8 views

Analytical expression for a two-dimensional saturation function

I need to define a saturation map on a two-dimensional space, which is further restricted by an additional constraint. More precisely, I am interested in determine a (possibly parameteric) analytical ...
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1answer
35 views

Why does this being a finite set mean that it cannot be the case that all element of the sequence are distinct?

I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Chapter 1.3 Data Flow Analysis says the following: The least solution. The ...
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2answers
53 views

Question of calculus concepts

Today I learned some concepts of increasing and decreasing functions, but some is so vague, so I have many questions. Here I can have a list of questions: Functions f(x) is decreasing on an interval (...

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