All Questions
1,375
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 ...
- 529
0
votes
1
answer
61
views
How to show that the function: $ f: (0,1) \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective?
How to show that the function: $ f: (0,1) \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective?
To prove this, I was thinking to start by supposing that $f(x_1)=f(x_2)$; this implies that ...
- 1,211
0
votes
1
answer
70
views
Prove that a function between two topological spaces is a homeomorphism
Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ?
My question is about the above, somebody has defined a function and its fine. But for ...
- 4,624
2
votes
2
answers
228
views
Extension of a differentiable function $f$ to an open superset
This is a question the book Munkres-Calculus on Manifolds pg.144(Exercise 3-b)
If $f :S\to \mathbb R$ and $f$ is differentiable of class $C^r$ at each point $x_0$ of $S$,then $f$ may be extended to a $...
- 1,138
1
vote
1
answer
23
views
Help to write the domain in a forma correct way
The question is simple: the domain of $f(x, y) = \ln(\sqrt{xy} + 1)$.
Now this is just $xy \geq 0$, which means either $x \geq 0$ and $y \geq 0$ or $y \leq 0$ and $y \leq 0$. This is easy to say and ...
- 157k
2
votes
2
answers
128
views
How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$
I want to evaluate this double integral:
$$
\iint \frac{y}{x} \, dx \, dy \quad
$$
which is bounded by functions:
$$
y = 0 \quad
$$
$$
y = x \quad
$$
$$
x^2 + 4y^2 = 4 \quad
$$
And is in the first ...
- 8,455
3
votes
2
answers
82
views
Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$
Let $f:[0,1]\to \mathbb R$ be a continous function on $[0,1]$ and differentiable on $(0,1)$, $f(0)=0, f(1)=1$. Prove that there exists $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where $e$ is the ...
0
votes
0
answers
54
views
Customizing the bump function
I have the standard bump function below.
$$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$
How can I customize it to be like below:
Translate and scale
I can translate and scale by:
$$ \Psi(x) ...
- 271
0
votes
1
answer
1k
views
When right inverse of a surjective mapping is continuous?
It is proved that if $(X,d)$ is a metric space and $f: X\to X$ is surjective, there exists $f^\ast:X\to X$ such that $f\circ f^\ast x=x$ for all $x \in X$. Here, $f^\ast$ is called right inverse.
I ...
CommunityBot
- 1
1
vote
1
answer
19
views
Determine if a vector valued function has a root based on the behaviour on the boundary
Say $f: \mathbb R^n \to \mathbb R^n$ is a vector valued continuous function. Say we further know that $t^\top f(t) \ge 0$ $\,\forall \,\,||t||=1$. Does this imply that $f$ has a root in the unit ball.
...
- 32.7k
0
votes
0
answers
18
views
Functions of bounded variation - inequality
Let $u: \mathbb{R} \to \mathbb{R}$ such that $u \in C^1 (\mathbb{R}) \cap \mathrm{BV} (\mathbb{R})$. Prove that
$$\frac{1}{\varepsilon} \int_{-\infty}^{\infty} |u(x+\varepsilon) - u (x)| dx \leq TV (u)...
- 31
1
vote
0
answers
40
views
Parametric prolate epicycloid modelling and integration
I was trying to model an epicycloid for my math assignment but none of the parametric equations I found ended up helping me model it on desmos.
One of the more prominent equations I found on the ...
- 1,512
1
vote
1
answer
32
views
Searching functions that satisfy the differential inequalities
I'm trying to find two nonnegative continuous functions $f_1$ and $f_2$ defined on a bounded domain $[0,a]$ such that satisfies the next conditions:
\begin{align}
f_1(x) > & f_2(x),\quad\text{...
- 21.6k
5
votes
3
answers
21k
views
Increasing/Decreasing intervals of a parabola
I am being told to find the intervals on which the function is increasing or decreasing.
It is a normal positive parabola with the vertex at $(3,0).$ The equation could be $y = (x-3)^2,$ but my ...
- 1,512
1
vote
2
answers
72
views
Confusion about two branches of Lambert $W$ function
I am confused about the usage of the two branches $W_{0}$ and $W_{-1}$ of Lambert $W$ function. Suppose $\epsilon > 0$ and we have an equation for the form:
$$-xe^{-x} = -e^{-1-\epsilon}$$
Note ...
- 7,212
0
votes
0
answers
184
views
Show that there is a $c$, such that $f'(c)=0$
Given that $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous and differentiable function satisfying
$$\int_a^b x^2f(x)dx ≠ \left(\frac{b ^ 3}{3} - \frac{a ^ 3}{3}\right)\cdot f(c) $$
for any $a, b \...
- 19
4
votes
0
answers
115
views
Show that $f(a,b,c)=(a+b+c)^3+(a+b)^2+a$ is injective.
For a function $f : \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by
$f(a,b,c)=(a+b+c)^3+(a+b)^2+a$
I want to show that $f$ is injective.
How can I show this?
I ...
- 41
0
votes
1
answer
59
views
Real Analysis - discontinuity
Let $f(x)$ = $x^2$ if $x$ is rational and $f(x)$ = $0$ if $x$ is irrational.
Prove that that $x$ is discontinuous at all $x\neq 0$
We look at the case where $x\neq 0$ and $x\in R\setminus Q$
Proof
Let ...
- 150k
17
votes
3
answers
366
views
How do I construct a function $\operatorname{sog}$ such that $\operatorname{sog}\circ\operatorname{sog} = \log$?
Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that
$$\sog(\sog(x)) = \log(x)$$
for all real $x>0$.
My questions:
Does such a function exist?
...
- 9,261
0
votes
0
answers
17
views
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 ...
- 3,482
1
vote
3
answers
118
views
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
Our teacher showed us this, and asked us to solve the rest. But I'm a bit confused by what he did.
$...
- 29.8k