All Questions
1,376
questions
2
votes
0
answers
47
views
Alternate proof to the Extreme Value Theorem
I'm following Spivak's Calculus and was revisiting some of my notes when I think I found a much more straightforward proof for the Extreme Value Theorem, compared to the one given in the book. I was ...
- 283
3
votes
2
answers
2k
views
Can the sigmoid function approximate any function (or relation) where $0<y<1$
I'm studying Machine Learning and Artificial Neural Networks. Some basic principles of Machine Learning are linear regression, multivariate linear regression, and nonlinear regression. The last of ...
CommunityBot
- 1
0
votes
1
answer
54
views
Prove that assuming $f:S\rightarrow T$, $f$ is a bijection iff there is $g:T\rightarrow S$ such that $f\circ g$ and $g\circ f$ are identity maps
I'm trying to prove the following:
Let $S$ and $T$ be sets and $f: S \rightarrow T$. Show that $f$ is a bijection iff there is a mapping $g: T \rightarrow S$ such that $f \circ g$ and $g \circ f$ are ...
0
votes
2
answers
45
views
Find the domain of this function through analytical ways
Consider $f(x) = \ln(2 + xe^{x^2})$ in the domain $(a, 0]$ where of course $a < 0$.
I was wondering if it's possible to find $a$ via analytical methods, using theorems and definition of analysis in ...
- 7,792
8
votes
4
answers
706
views
How to Find Efficient Algorithms for Mathematical Functions?
Context: I had to write a code that would compute $\arctan(x)$ for all real $x$ with an error less than $10^{-6}$. The only algorithm I could think of was using the Taylor series of $\arctan(x)$, ...
- 268k
0
votes
1
answer
2k
views
What type of function is this (derivative of a hyperbola)?
The derivative of the hyperbola $$f(x)=\frac{b}{a}\sqrt {a^2+x^2}$$
is
$$f'(x)=\frac{bx}{a\sqrt {a^2+x^2}}$$
The graph (for $a=b=1$) looks somewhat like a Sigmoid function, but I honestly cannot ...
CommunityBot
- 1
4
votes
4
answers
730
views
How to find a Newton-like approximation for that function?
I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$.
added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no ...
- 268k
0
votes
0
answers
12
views
Controlling size of image intervals from the derivative
I am trying to understand the proof of van der Corput's inequality on exponential sums. Basically, we assume that we have a twice continuously derivable function $f$ on a bounded interval $I$ such ...
- 1,285
1
vote
0
answers
42
views
Locally Lipschitz function and continuity
In my book, when It comes to prove that the integral function Is continuos on an interval X, It shows that it's "locally Lipschitz" on X and, therefore, continuos.
At a First read, I didn't ...
- 33
0
votes
0
answers
11
views
Approximation a piecewise affine function with sigmoid function
I am dealing with the following piecewise affine function:
$$
f(x) =
\begin{cases}
0, & \text{if } x \in [0,1] \\
x - 1, & \text{if } x \in [1,2] \\
1, & \text{if } x \in [2,3] \\
\end{...
- 109
1
vote
1
answer
142
views
Finding a non-affine function satisfying symmetry properties
I am looking for an example of a continuous, non-affine function $u\colon X\to \mathbb{R}$ and a continuous, non-negative function $\epsilon\colon X\to\mathbb{R}_{\geq 0}$ such that the following hold ...
- 93.3k
0
votes
1
answer
1k
views
continuous extension and smooth extension of a function
Let $X$ be a metric space. Let $E$ be a subset of $X$.
(1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such ...
CommunityBot
- 1
6
votes
2
answers
298
views
Examples of continuous functions that are monotone along all lines
I am looking for different examples (or even a complete characterization if this is possible) of continuous functions that are monotone along all lines. By that I mean functions $f\colon X\to\mathbb{R}...
- 4,401
2
votes
2
answers
1k
views
A continuous onto/surjective function from $[0, 1) \to \Bbb R$.
Does there exist a continuous onto/surjective function from $[0, 1) \to \Bbb R$?
Finding difficult to site an example...
- 483
1
vote
0
answers
17
views
Set valued approximate inversion
I have a function $f: \mathcal{D} \to D$ where $\mathcal{D}$ is some domain of interest.
Now let function $g_\theta: \mathcal{D} \to \mathcal{P}(\mathcal{D})$ be a set valued function, Here $\mathcal{...
- 2,829
0
votes
2
answers
466
views
Approximation of Sobolev Function in $W^{k,p}(\Omega)$ by Continuous Functions of limited Regularity
Is $C^{k+1}(\Omega)\cap W^{k,p}(\Omega) $ dense in $W^{k,p} (\Omega)$? Assuming $\Omega$ has a $C^1$ boundary (or Lipschitz continuous boundary, if allowed?). I know the standard results on smooth ...
0
votes
1
answer
36
views
Upper bound of $2\vert\cos{\frac{x+y}2}\sin{\frac{x-y}2}\vert$
We have $2\left|\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\right|$
As $\left|\cos\left(\frac{x+y}{2}\right)\right|\leq 1$
$\left|\sin\left(\frac{x-y}{2}\right)\right|\leq 1$
Is it ...
- 851
1
vote
1
answer
36
views
Can a non-constant continuous function be constant on these hyperbolas?
Can a non-constant continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following hyperbolas?
$$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$
$$H_\infty=\mathbb{R}\times\{0\}$$
...
- 32.6k
2
votes
1
answer
1k
views
Ratio of convex functions with dominating derivatives is convex?
Let $f,g:\mathbb [0,\infty)\rightarrow (0,\infty)$ satisfy $f^{(n)}(x)\geq g^{(n)}(x)>0$ for all $n=0,1,2,\ldots$ and $x\in [0,\infty)$. In particular, $f\geq g> 0$ are increasing and convex (...
CommunityBot
- 1
0
votes
1
answer
38
views
Constructing a Continuous Function Below an Increasing Function
Let $f$ be an increasing function defined on $[0,1]$ with $f(0)=0$ and $f(x)>0$ for $x>0$. Does there exists a continuous function $g$ on $[0,1]$ such that $g(x)>0$ on $(0,1]$ and
$$g(x)\leq ...
- 368
20
votes
7
answers
26k
views
$f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$
Find a function $f(x)$ such that:
$$f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$$
with $f(4)=65$.
I have tried to let $f(x)$ be a general polynomial:
$$a_0+a_1x+a_2x^2+\ldots a_nx^n$$
which leaves $f(\frac{...
- 2,792
3
votes
0
answers
87
views
Find the values of $b$ for which $f(x)=x^3+bx^2+3x+\sin(x)$ is bijective
Find the values of $b$ for which $f(x)=x^3+bx^2+3x+\sin(x)$ is bijective.
As we know $f(x)$ is surjective, the only task left to prove it bijective is to prove that $f(x)$ is strictly monotonic (...
- 50.3k
2
votes
0
answers
49
views
Prove that $(f+g)(x)<t$ ($t\in\mathbb{R}$) holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I got stuck on this question:
Prove that $(f+g)(x)<t$, $t\in\mathbb{R}$, holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I think one direction is ...
- 2,493
3
votes
2
answers
74
views
Prove that $g(x) = \sum_{n=0}^{+\infty}\frac{1}{2^n+x^2}$ ($x\in\mathbb{R}$) is differentiable and check whether $g'(x)$ is continuous.
The function $g(x)$ is a function series, so it is differentiable when $g'(x)$ converges uniformly. So I should just check uniform convergence of $g'(x)$ by using the Weierstrass M-test:
$$g'(x) = \...
- 33.9k
0
votes
0
answers
60
views
How to prove that $f : [0,1] \to [0,1] \times [0,1]$ is continuous?
I'm trying to show that the function
$$ f : [0,1] \to [0,1] \times [0,1] $$
$$ t=0.t_1 t_2 t_3 \dots \mapsto (0.t_1 t_3 t_5 \dots, 0.t_2 t_4 t_6 \dots ) $$
is continuous. My idea was to show that the ...
- 61
-1
votes
2
answers
117
views
Need help with creation of an example [closed]
I've been struggling for days now and I cannot come up with an example of a function $f(x)$ that satisfies the following:
$|f(x)−f(y)| < |x−y|$ for any $x, y ∈ R$
AND
equation $f(x) = x$ does ...
- 9
-2
votes
1
answer
41
views
About the Exponential function
Consider function $y=a^x$, $a>1$, and we need to show that
$$\frac{2(a-1)}{(a+1)} < \ln(a) < -1 + \sqrt{2a-1}$$
My idea is to use $y=a^x = \exp{(x\ln a)}$, then find the derivative of $y$ at $...
- 1,131
2
votes
3
answers
343
views
Question about definition of Sequences in Analysis I by Tao.
Here's the definition of a sequence as laid out in the text:
Let $m$ be an integer. A sequence $(a_n)_{n=m}^\infty$ of rational
numbers is any function from the set $\{n \in \mathbf{Z} : n \geq m\}$ ...
- 11.2k
2
votes
1
answer
48
views
Find all $f:\Bbb R\to\Bbb R$ st for any $x,y\in\mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ equals the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
Note: The ...
- 2,665
2
votes
1
answer
122
views
Topological version of uniform convergence of functions
We have a sequence of continuous functions $\{f_n\}$ on a Banach space $X$ and $f_n(x)\to f(x)$ for each $x\in X$ as $n\to\infty$. Given an open ball $B\subset X$ and $\epsilon>0$, we want to show ...
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