All Questions
1,374
questions
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 ...
8
votes
4
answers
692
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 $\arctan(x)$, ...
- 6,332
0
votes
0
answers
11
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
41
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
9
views
Approximation a piecewise affine function with sigmoid function
I am dealing with the following piecewise affine function:
$$
f(x) =
\begin{cases}
0, & \text{if } x \in [0,1] \\
x - 1, & \text{if } x \in [1,2] \\
1, & \text{if } x \in [2,3] \\
\end{...
- 107
6
votes
2
answers
295
views
Examples of continuous functions that are monotone along all lines
I am looking for different examples (or even a complete characterization if this is possible) of continuous functions that are monotone along all lines. By that I mean functions $f\colon X\to\mathbb{R}...
- 356
1
vote
1
answer
142
views
Finding a non-affine function satisfying symmetry properties
I am looking for an example of a continuous, non-affine function $u\colon X\to \mathbb{R}$ and a continuous, non-negative function $\epsilon\colon X\to\mathbb{R}_{\geq 0}$ such that the following hold ...
- 356
1
vote
0
answers
17
views
Set valued approximate inversion
I have a function $f: \mathcal{D} \to D$ where $\mathcal{D}$ is some domain of interest.
Now let function $g_\theta: \mathcal{D} \to \mathcal{P}(\mathcal{D})$ be a set valued function, Here $\mathcal{...
- 2,819
0
votes
1
answer
36
views
Upper bound of $2\vert\cos{\frac{x+y}2}\sin{\frac{x-y}2}\vert$
We have $2\left|\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\right|$
As $\left|\cos\left(\frac{x+y}{2}\right)\right|\leq 1$
$\left|\sin\left(\frac{x-y}{2}\right)\right|\leq 1$
Is it ...
- 35
0
votes
1
answer
36
views
Constructing a Continuous Function Below an Increasing Function
Let $f$ be an increasing function defined on $[0,1]$ with $f(0)=0$ and $f(x)>0$ for $x>0$. Does there exists a continuous function $g$ on $[0,1]$ such that $g(x)>0$ on $(0,1]$ and
$$g(x)\leq ...
- 319
3
votes
0
answers
86
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 (...
- 487
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 ...
- 2,473
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?
$$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$
$$H_\infty=\mathbb{R}\times\{0\}$$
...
- 311
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:
$$g'(x) = \...
- 131
0
votes
0
answers
58
views
How to prove that $f : [0,1] \to [0,1] \times [0,1]$ is continuous?
I'm trying to show that the function
$$ 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