All Questions
1,376
questions
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53
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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
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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 ...
- 7,814
8
votes
4
answers
698
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 $\arctan(x)$, ...
- 267k
0
votes
1
answer
2k
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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 ...
- 267k
0
votes
0
answers
11
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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
41
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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
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votes
0
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9
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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{...
- 107
1
vote
1
answer
142
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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 ...
- 93.1k
0
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1
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1k
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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
295
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}...
- 4,381
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...
- 491
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{...
- 2,819
0
votes
2
answers
453
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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
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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 ...
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