All Questions
Tagged with supremum-and-infimum continuity
138 questions
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151
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Prove $\sup_{P=(t_0,\dots,t_k)}\sum_{j=1}^k |f(t_j)-f(t_{j-1})|=\sup_{n\in\mathbb N}\sum_{j=1}^{k_n} |f(t_j^n)-f(t_{j-1}^n)|$
Let $f:[0,1]\rightarrow \mathbb R$ be right continuous $P_n=(0=t_0^n<t_1^n<\dots<t_{k_n}^n=1)$ a sequence of grids with $\max_{1\leq j\leq k_n} |t_j^n-t_{j-1}^n|\to 0 (n\to \infty)$. Show ...
- 153
2
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1
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76
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Finding the different zeros of a continuous function
I'm working on Spivak's Calculus and am doing Problem 4 of Chapter 8. Here's the problem:
Suppose $f$ is continuous on $[a,b]$ and that $f(a) = f(b) = 0$. Suppose also that $f(x_0) > 0$ for some $...
- 353
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0
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50
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Show that the supremum of a family of continuous functions is lower semicontinuous.
Let $(X, \mathcal{T})$ be a topological space. Consider a family $\{f_i \mid i \in I\}$ of continuous functions $f_i : X \rightarrow \overline{\mathbb{R}}.$ Define $M : X \rightarrow \overline{\mathbb{...
- 93
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19
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IF $\Psi(x) = \inf_{0 \leq s \leq t} \Phi_s(x)$,can we have $\Psi^{-1}(x) = \sup_{0 \leq s \leq t} \Phi_s^{-1}(x)$?
{$\Phi_s(x)$}$0 \leq s \leq t$ is a set of continuous and increasing functions which are uniformly bounded by $[0,1]$.Defind $\Psi(x) = \inf_{0 \leq s \leq t} \Phi_s(x)$,can we have $\Psi^{-1}(x) = \...
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51
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obtain the supremum and infimum of the given set.
Consider the set $E=\{cos \frac{n\pi}{3}+\sin \frac{n\pi}{3}:n\in \mathbb{N}\}$. Then supremum of E and infimum of E is
(a) 1 and -1
(b) $\frac{\sqrt{3}}{2}+\frac{1}{2}$ and $\frac{1}{2}-\frac{\sqrt{...
- 1,297
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0
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106
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What is $\sup\limits_{f \in \mathscr F} \int_{1}^{3} \frac {f(x)} {x}\ dx\ $?
Let $\mathscr F$ be the set of all continuous functions $f : [1,3] \longrightarrow [-1,1]$ such that $\int_{1}^{3} f(x)\ dx = 0.$ Find the value of $$\sup\limits_{f \in \mathscr F} \int_{1}^{3} \frac {...
- 566
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Find all the values that the expresion $\sum_{cyc} \frac{a}{a+b+d}$ takes.
Let $a,b,c,d \in \mathbb{R}^+$. Find all the values that the expresion $\frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}$ takes.
$$\\$$ I had this problem in a test. I found that $...
- 362
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2
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268
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Is it possible to prove that $|\sup (f+g)| \le |\sup f| + |\sup g|$? [duplicate]
Let C the space of continuous functions $f: [0,1] \to \mathbb R$. How to prove that $|\sup (f+g)| \le |\sup f| + |\sup g|$?
I thought that it was a simple consequence of the well-known fact that $\...
- 59
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393
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How can we define the supremum norm on the space of continuous functions if they're not necessarily bounded?
I frequently see the uniform norm described as a norm on the space of continuous functions $C(Y, X)$ where $Y$ is a topological space and $X$ is a metric space, especially in the context of the ...
- 257
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Is my little proof of lub axiom using (lub axiom in any way)?
Below is my little proof of lub axiom
DEFINITIONS :
$A$ be any non empty subset of $\mathbb{R}$ bounded above
$B$ be the set of all real upper bounds of $A$
$x \in B\ \Leftrightarrow \forall\ a \in A,...
- 556
1
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1
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289
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Suppose function $f(x)$ is continuous on $[a,b]$ then the function $\sup_{a\le{t}\le{x}}f(t)$ is also continuous.
In this problem, I need to prove $\sup_{{a}\le{t}\le{x}}f(t)$ is continuous, and I denote this function by $M(x)$. It is obvious that the function $M(x)$ is a non-decreasing function，so I can conclude ...
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417
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Are inf and sup continuous functionals in general?
Let $X$ be any topological space and $\bar{\mathbb{R}} = [-\infty, \infty]$ with the standard topology. Is it true, in general, that the functionals $$\inf: C(X,\bar{\mathbb{R}}) \to \bar{\mathbb{R}}, ...
- 4,559
5
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107
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maximum Fourier coefficient among all bounded real continuous periodic functions
Let $\mathcal C$ be the space of real continuous $2\pi$-periodic functions bounded in absolute value by 1.
Can we compute
$$c_s := \sup_{f\in\mathcal C} \left|\int_0^{2\pi} e^{-isx}f(x)dx\right|$$
for ...
- 11.8k
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157
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Running supremum of a right-continuous function: prove that $\left\{x\geq 0\left|\sup_{0\leq y\leq x}f(y)\geq K\right.\right\}$ is closed.
Let $f:[0,\infty)\rightarrow\mathbb{R}$ be a right-continuous function. Consider the set
$$
E=\left\{x\geq 0\left|\sup_{0\leq y\leq x}f(y)\geq K\right.\right\},
$$
where $K$ is a constant. It is ...
- 1,352
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36
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Maximal inequality for time-continuous martingales: counter-example
The following can be regarded as the generalization of the maximal inequality for time-continuous martingales:
Let$ (X_t)_{t \ge 0}$ be a supermartingale with right-continuous sample paths. Then, for ...
- 124
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44
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Proving that for $f\in C^2(a,\infty)$ then $\max_{(a,\infty)} |f'(x) |\le 4\max_{(a,\infty)}|f(x)|\max_{(a,\infty)} |f''(x) |$. [duplicate]
Let $f\in C^2(a,\infty)$
and $M_k=\max_{(a,\infty)} |f^{(k)}(x) |$, $k=0,1,2$, prove that
$$M_1^2\le 4M_0M_2.$$
Firstly, I know that if $f, g\in C^2(a,\infty)$ so is $fg\in C^2(a,\infty)$.
Any hint on ...
- 401
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1
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133
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I want to prove $f$ in continuous on $[a,b]$ implies $F(x)=\sup f([a,x])$ is continuous on $[a,b]$?
Goal: I want to prove $f$ in continuous on $[a,b]$ implies $F(x)=\sup f([a,x])$ is continuous on $[a,b]$.
Proof:
Suppose $f$ in continuous on $[a,b]$.
Let $c \in (a,b)$. Let $\epsilon>0$.
By the ...
user1104987
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83
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The measurability of $c$-transform
In my proof of below result, I'm surprised that $f$ has no effect on the measurability of $f^c$.
Could you have a check on my attempt?
Is there stronger assumption that makes $f^c$ continuous?
Let $...
- 5,991
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1
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53
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Prove that $\bar{B}_1(0)=\{f \in \mathcal{C}^{1}([0,1]) | \: \lVert f \rVert_{1,\infty} \leq 1\}$ is relatively compact but not compact [closed]
Define on $\mathcal{C}^{1}([0,1])$ the norm $\lVert f \rVert_{1,\infty} := \lVert f\rVert_{\infty} + \lVert f' \rVert_{\infty}$.
Prove that the closed unit ball $\bar{B}_1(0) \subset \mathcal{C}^{1}([...
- 569
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47
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The supremum function for ideals is continuous
Let $P$ be a dcpo, $Idl(P)$ be the set of ideals ordered by inclusion, and $f : Idl(P) \to P$ be the function defined by $f(I) = \bigvee I$.
Question: How to show that $f$ is Scott-continuous?
My ...
- 1,578
1
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1
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102
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A multiple choice question on a function
Let $f(x) = x^2+ {1\over x^2}$ for $x \in (0,\infty)$, then which of the following is/are correct:
(a) $f$ is continuous on $(0,\infty)$.
(b) $f$ is uniformly continuous on $(0,\infty)$.
(c) $f$ ...
- 940
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30
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Hatcher K-Theory continuity of $\alpha \to \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$
Does Hatcher makes an error at page $45$ afirming that $f:\alpha \mapsto \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$ is continuous? I didn't find any reference of continuity of $\inf$ of ...
- 5,204
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1
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49
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Show $\inf\{t\ge0:x(t-)\in B\text{ or }x(t)\in B\}\le T$ iff $x(T)\in B=\bigcap_nB_n$ or $\forall n:\exists s\in[0,T):x(s)\in B_n$
Let $(E,d)$ be a metric space, $x:[0,\infty)\to E$ be càdlàg (right-continuous with left-limits), $x(0-):=x(0)$, $x(t-):=\lim_{s\to t-}x(s)$ for $t>0$, $B\subseteq E$ be nonempty and $$\tau:=\inf\...
- 13.9k
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54
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Why is d' a metric in this case?
If $X$ is a compact space and $Y$ a metric topological space with metric $d$, then $d'(\alpha,\beta) = sup_{x \in X} d(\alpha (x),\beta (x))$ is a metric on $TOP(X,Y)$
$TOP(X,Y)$ denotes the set of ...
- 306
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75
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Understanding the proof of an extension theorem
I'm studying the proof of the following theorem, but there are two things that I have not been able to understand about the continuity of the function $g$ on the boundary of $A$.
(1) Why $\inf_{y \...
- 551
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0
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33
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Calculus on the half positive space
Let $n \geq 2$ be an integer, and $J: \mathbf{R}^{n} \longrightarrow \mathbf{R}$ a continuous, coercive and strictly convex fonction.
For any real $\delta \geq 0$, we denote by $U_{\delta}=\mathbf{R}^{...
- 183
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149
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Proving that $\sup_{s \in [a,b)}f(s)=\sup_{s \in [a,b)\cap \mathbb{Q}}f(s)$ for right continuous function.
Proposition. Let $[a,b) \subset \mathbb{R},\,a<b$. If $f:[a,b)\to \mathbb{R}$ is right-continuous over $[a,b)$, we have that
$$\sup_{s \in [a,b)}f(s)=\sup_{s \in [a,b)\cap \mathbb{Q}}f(s)$$
This ...
- 17.1k
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1
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33
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Continuity in different metrics:
Let $X = C([0,1];\mathbb{R} )$ be space of continuous real valued fns defined on [0,1] equipped with sup metric $d$ given by : $$d(f,g) = sup_{x \in [0,1]} |f(x) - g(x) |$$
Consider map $I: X \...
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If $u$ is rational, can we show $\inf\{t:x(t)\in B\}\le u$ iff $x(s)\in B$ for some rational $s\le u$?
Let $(E,d)$ be a metric space, $B\subseteq E$ be closed and nomepty, $x:[0,\infty)\to E$, $I:=\{t\ge0:x(t)\in B\}$ and $\tau:=\inf I$. If $I$ is nonempty and $\tau\in I$, then we easily see$^1$ that $$...
- 13.9k
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$a^*$ and $a^{**}$ in wikipedia's proof of IVT
I am following the proof of the IVT on Wikipedia, and have one point of confusion. In the proof they say
By the properties of the supremum, there exists some $a^* \in (c-\delta, c]$ that is contained ...
- 6,641
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208
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Fundamental theorem of calculus for monotone function
Assume $F:(0,1] \to \mathbb{R}$ is given by $F(x) = \int_0^xf(u) du$ where $f:(0,1) \to \mathbb{R}$ is a nonincreasing (not necessarily continuous) function (note that F is real-valued so it is ...
- 928
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15
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Show that $\limsup_{t\to T_0-}\left\|u(t,\;\cdot\;)\right\|_\infty<\infty$ is not possible here
Let $T\in(0,\infty]$ and $\Omega\subseteq\mathbb R^d$ be bounded and open. For $k\in\mathbb N$, let $I_k:=(0,\min(k,T))$, $u_k\in C(\overline{I_k}\times\overline\Omega)$ and $$T_k:=\min\left(\inf\left\...
- 13.9k
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2
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182
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Prove that $\sup_A f = \sup f|_{\mathbb Q\cap A} = \sup f |_{(\mathbb{R} \setminus \mathbb{Q})\cap A}$ [closed]
I was trying to prove something else and this came up. It must be really simple to prove but I am struggling to do it. Let $A$ be a non-empty and compact interval of $\mathbb{R}$ and $f \colon A \to \...
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1
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53
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If $f$ is continuous, $\sup_{|\lambda|<\omega} |f(\lambda)|=\infty$ then $\lim_{|\lambda| \rightarrow \omega} |f(\lambda)|=\infty$
Suppose that $f:(-\omega,\omega) \rightarrow \mathbb{R}$ is a continuous function, $\omega>0$ and
$$\sup_{|\lambda|<\omega} |f(\lambda)|=\infty.$$
Can we conclude that
$$\lim_{|\lambda| \...
- 2,413
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2
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65
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If $M$ is compact, then follows from $\forall x \in M | f(x) < a$ that $\sup_{x\in M} f(x)<a$
Let $(X,d)$ be a metric space, $M\subseteq X$ and $f: M \to \mathbb{R}$ continuous. Show that:
If $M$ is compact, then follows from $\forall x \in M | f(x) < a$ that
$\sup_{x\in M} f(x)<a$.
My ...
- 197
0
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1
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427
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Maximum vs supremum of a discontinuous function over a closed bounded set
The statement "Any continuous function must have a maximum on a closed bounded set" is made in these notes.
We are looking at a function $f: S\rightarrow \mathbb{R}$. I can see why ...
- 1,758
0
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1
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400
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Continuity of a supremum function over compact sets
I'm unsure how to formally prove or disprove the following claim. It came up in trying to prove convergence of a gradient descent-style algorithm. I've tried using different definitions of continuity ...
- 1,824
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1
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69
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Picard-Lindelöf to show whether $u'=u^2-u^3$ has unique solution on an interval
This problem is taken from here.
Consider the initial value problem:
$$
u'=f(u,t)=u^2-u^3$$
$$ u(0)= 2/a>0 $$
$a$ is a small constant.
How can I determine wheter a unique solution tom $t=0$ to $t=...
- 159
0
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1
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126
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Supremum and Continuity for function
Let $f$ be continuous on $[a, b]$. Define a function $g$ as follows: $g(a)=f(a)$ and, for $x$
in $(a, b]$ $$g(x)=\sup \{f(y): y \text { in }[a, x]\}$$
Prove that $g$ is monotone increasing and ...
- 1,023
-1
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1
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29
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If $\tau:=\{t\ge t_0:f(t)\in B\}$, can we show that $f(\min(t,\tau))\not\in B$ for all $t$?
Let $t_0\in\mathbb R$, $f:[t_0,\infty)\to\mathbb R$ be continuous, $B\subseteq\mathbb R$, $I:=\{t\ge t_0:f(t)\in B\}$, $\tau:=\inf I$ and $$g(t):=f(\min(t,\tau))\;\;\;\text{for }t\ge t_0.$$
If$^1$ $f(...
- 13.9k
0
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2
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85
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continuity, supremum/infimum
I came across this but I'm still struggling to understand the concept fully. Specifically, how do we know $\text{lub}\{f(x)\mid a-\epsilon<x<a+\epsilon\}-\text{glb}\{f(x)\mid a-\epsilon<x<...
- 349
1
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1
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82
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Continuity of Functions defined from a supremum
Let $\overline{\Omega}\subset\mathbb{R}^{N}$ be a compact set and $T<\infty$. Assume $f : \overline{\Omega}\times(0,T)\to\mathbb{R}$ is continuous and sastifies
(A) $\forall x\in\overline{\Omega}, \...
- 1,316
1
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1
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43
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Continous functions and supremum
this might be a very trivial questions that can be answered by pointing to a theorem or lemma, but I have trouble remembering something clearly. Give a continuous function $f(x)$ with $x\in \mathbb{R}^...
- 1,652
1
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1
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52
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Finding infimum and proving continuity of a function
Let $f : [a,b] \to\mathbb R$ and for every $x \in [a,b]$ there exists $y \in [a,b]$ such that $|f(y)| < \frac12|f(x)|$.
Find $\inf\{|f (x)| : x \in [a, b]\}$, and show that $f$ is not continuous on ...
0
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1
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169
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Is $\sup\{ \max\{f(x)\}\} = \sup\{f(x)\}$?
In some problem from my differential equations course, I stated that, given $f(x)$ a continuous function defined in a compact set $K$,
$$\sup_{x\in K} \{ \max_{x\in K}\{f(x)\}\} = \sup_{x\in K}\{f(x)\}...
- 3,609
3
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2
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299
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Spivak Rising Sun Lemma
Suppose that all points of $(a,b)$ are shadow points, that is $\forall x \in (a,b) \space \exists y\space(y>x \mbox{ and } f(y)>f(x))$.
For all $x\in(a,b)$, prove that $f(x)\leq f(b)$.
let $A=\{...
- 726
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1
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129
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On Suprema and limits
Let be $f:I\to\mathbb{R}$ a continuous function, $[x,x_0]\subseteq I$ and $M:=\sup\{f(y)\mid y\in[x,x_0]\}$. Show that the supremum attains the value of $f(x_0)$ when $x\to x_0$.
My original idea was ...
- 4,799
0
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0
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81
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Prove the following for the monotonic function $f$ on $[0,1]$ satisfying $f \left (\frac {1} {4} \right ) f \left (\frac {3} {4} \right ) \lt 0.$
If $f : [0,1] \longrightarrow \Bbb R$ be a monotonic function with $f \left (\frac {1} {4} \right ) f \left (\frac {3} {4} \right ) \lt 0.$ Let $\sup \{x \in [0,1]\ \big |\ f(x) \lt 0 \} = \alpha.$ ...
- 2,932
1
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2
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63
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Show that the following is uniformly convergent
Show that if the function $f: R \to R$ is uniformly continuous, then the sequence $f_{n}(x) = f(x + \frac{1}{n})$ converges uniformly.
I know that f(x + 1/n) approaches f(x) as $n \to \infty$. I feel ...
- 1,136
-1
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2
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26
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Could I define the supremum of a function, given that that function is not continuous?
Starting from a function $f(x)$ which is not continuous on $\mathbb{R}$, it that possible to "define" objects like $\sup\limits_x f(x)$, $x\in\mathbb{R}$?
