# 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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### Knowing $f(t^\ast)\ge 0$ (and some other information), can we show that $f(t)\ge 0$ at $t<t^\ast$?

I asked a similar question before and had to make several changes so before anyone spends time on answering it, I decided to clarify here. We have constants $\alpha_1,\alpha_2,r_1,r_2,c>0$ and we ...
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### How many monotonic functions on sets of naturals are there?

A function $f\colon P(\mathbb{N}) \to P(\mathbb{N})$ is monotonic iff $x \subseteq y \implies f(x) \subseteq f(y)$. What is the cardinality of the set $F$ of all such functions? What I have tried: I ...
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### Function derivative and monotonicity

Why do we say f'(x) >= 0 for all x in the interior of I exactly if f(x) is increasing on all of I; Instead of f'(x) >= 0 for all x in the interior of I exactly if f(x) is nondecreasing on all ...
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### Monotonically decreasing or increasing sequence

Let ${x_n}$ be a sequence of real numbers. consider the set $$P=\{n\in \mathbb{N}:x_n>x_m \forall m \in \mathbb{N}\hspace{4 pt} with \hspace{4 pt}m>n \}$$ Then which of the following is/are ...
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### Does this function is monotonically incresing about a?

Suppose that $\boldsymbol{t}\sim \mathcal{N}(\boldsymbol{u};\boldsymbol{0},\boldsymbol{M})=f_{\boldsymbol{t}}(\boldsymbol{u})$, where $\boldsymbol{t}$ is a $N$-dimensional gaussian random vector, and \...
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### How to solve for coefficients of monotonically increasing polynomial that must pass through two states?

Preface: I posted a similar question yesterday but wanted to change it slightly. I have a problem I can solve, but I want to modify it slightly and I'm unsure of the best way to solve the new problem. ...
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Here is the setup: Let f be a non-negative measurable function on $\mathbb{R}$ where $\int_\mathbb{R}f\text{d}m_1=2021$. Show that the limit exists and compute $\lim_{n\rightarrow \infty}\int n\log(1+\... 1 vote 0 answers 58 views ### Integral of (generalized) inverse functions with a twist Consider a weakly increasing function$f : [0,1] \rightarrow [0,1]$and define its generalized inverse (since an inverse does not necessariliy exist unless it increases strictly) as $$f^{-1}(t):= \... -1 votes 1 answer 82 views ### Proving that given a bounded function f:\mathbb{R}\to\mathbb{R}, there exists a strictly increasing sequence (x_n) for which (f(x_n)) converges [closed] I don't know where to start with proving it. Thanks for the help. Let f:\mathbb{R}\to\mathbb{R} be a bounded function. Prove that there exists a strictly increasing sequence (x_n) such that the ... 3 votes 1 answer 71 views ### Show the existence of a sorting function Let (X,\leq) be a totally ordered set. A sort for f\in X^n is an element g\in X^n satisfying (i) g is nondecreasing. (ii) g=f\circ \sigma for some permutation \sigma:\{1,\dots,n\}\to \{1,\... 6 votes 3 answers 147 views ### Proving that a function f:[0,\infty)\to[0,\infty) satisfying these conditions is necessarily non-decreasing I have a function f: [0, \infty) \to [0, \infty) which is smooth. I also have that f(0) = 0 f'(0) > 0 f''(x) \leq 0, for all x \in [0, \infty) It intuitively makes sense then that f ... 0 votes 2 answers 35 views ### Asymptotically monotonic sequence Let (u_n)_{n \geq 1} be a sequence of real numbers such that$$u_n \sim c n$$where c>0. Is it true that (u_n)_{n \geq 1} is necessarily asymptotically increasing (i.e. that there exists an ... 0 votes 0 answers 17 views ### If a function to be maximized is almost everywhere differentiable can we say the derivative=0 condition must hold almost everywhere? We have a known, increasing function p:[0,1]\rightarrow[0,1]. We want to find conditions on functions R_1(\cdot),R_0(\cdot), R_1(\cdot) known to be strictly increasing and R_0(\cdot) known to ... 1 vote 0 answers 31 views ### Linear combination of "basis" to approximate any continuous monotone increasing functions Define$$ \mathcal{A} = \{ f(x) | x \in [a,b],f(x) \text{ is continuous monotone increasing} \}$\mathcal{A}$is not a vector space since there is no additive inverse. Is there a subset$\mathcal{A}...
A function $g:[a,b]→\mathbb{R}$ is increasing if $∀x_1{<}x_2$ in $[a,b]$, $g(x_1)≤g(x_2).\quad$ Similar is the definition of a decreasing function. So, if a function $f:[a,b]→\mathbb{R}$ satisfies ...