All Questions
Tagged with real-numbers functions
32
questions with no upvoted or accepted answers
3
votes
0
answers
77
views
$f(f(...f(x)...))$ $a$ times, where $a\in\mathbb{R}$
Take $f(x)$ and do a "double-call":
$f^2(x)=f(f(x))$
I use this notation here to explain my problem.
This can be easy calculated for any function. Also $f^{100}(x)$ is not really a problem. This ...
- 1,545
2
votes
0
answers
72
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Proving integral of a continuous function is continuous
Let $U \subset \mathbb{R}^N$ be an open set, let $f : U \times [a, b] \to \mathbb{R}$ be a continuous function. Consider the function $$g(x):= \int_a^b f(x,y) \,dy$$
with $x \in U$.
i) Prove ...
- 1,913
2
votes
0
answers
658
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Monotonically increasing function and Lipschitz continuous functions
Let $f(t,x):[0,T]\times \mathbb{R}\rightarrow \mathbb{R}$.
If $$|f(t,x)-f(t,y)|\leq C|x-y|, C>0$$ (Lipschitz continuous functions).
I need to found a monotonically increasing function $g(t;x)$ ...
- 73
1
vote
0
answers
43
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How to combine the $4$-dimensions of spacetime into 1 dimension?
I have been thinking about the possibility of representing all points in a $4$-dimensional spacetime coordinate system $\mathbb{R}^{1,4}$, as points on one line $P$ (or axis of a $1$-dimensional ...
- 126
1
vote
0
answers
149
views
What does trigonometric functions of real numbers really mean?
According to right triangle definition, trigonometric functions relates the angle to the ratio of sides of triangle. These functions take angles as input.
According to unit circle definition, ...
1
vote
0
answers
17
views
Looking for a sigmoid-like function with different properties
I am looking for a function that is 0 at 0, 1 at 1, increases to a predetermined $x_1$ relatively quickly, acquires a derivative close to 0 (but doesn't actually plateau) then starts increasing at $...
- 3,439
1
vote
0
answers
70
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Is this a surjection from $(0,1) \rightarrow \mathbb{R}$?
I am trying to think of creative bijections from $(0,1) \rightarrow \mathbb{R}$ that can serve as a proof of the fact that the cardinality of the interval $(0,1)$ is equal to the cardinality of $\...
- 1,186
1
vote
0
answers
94
views
Best approximation of a real number with two functions
There's two functions, called $F(x)$ and $G(x)$, where $F(x)>x,1<G(x) < x$ ,$F'(x)>0, G'(x)>0$ on $(2,\infty)$, and $F(x),G(x)\in(2,\infty)$. Given $x, y\in (2,\infty)$, and now I want ...
- 5,054
1
vote
0
answers
44
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Question concerning defining a particular class of functions
I have a multiset of real numbers $X \subseteq \mathbb{R} $ and I want to create a class of injective function to map the elements of $X$ to the unit interval(so basically a normalization).
However ...
- 75
1
vote
0
answers
23
views
Existence of a particular transformation
I've a set of data points $S = \{ x | x\in [0,1]\}$ (i.e. real values from the unit interval). In some cases I've big clusters in the data and I want to spread the values in between the unit interval ...
- 75
0
votes
1
answer
46
views
How can I study the changes of $f_k(x)=\frac{e^{kx}-1}{2e^x}$
Let $f_k(x)$ be a function defined on $\mathbb{R}$ by
$$f_k(x)=\frac{e^{kx}-1}{2e^x}$$
Where $k$ is a real , How can I study according to the values of $k$ the changes of the changes of $f_k$
I ...
- 59
0
votes
1
answer
26
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Is it possible for a function to be continuously derivable over its entire open domain except for a removable discontinuity?
For example, is there a function $f \in \mathcal{C}(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R} \setminus \{ 0 \})$ such that
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
- 33
0
votes
0
answers
30
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Complex Functions Examples
I was asked to give an example of a function:
i) whose domain isn't equal to its codomain
ii) whose domain isn't equal to its image
iii) whose codomain isn't equal to its image
iv) a function $f$ from ...
user1071088
0
votes
0
answers
124
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Converse of Bolzano Weierstrass Theorem
Bolzano Weierstrass Theorem (for sequences) states that Every bounded sequence has a limit point.
However the converse is not true i.e. there do exist unbounded sequence(s) having only one real limit ...
0
votes
0
answers
36
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Is there a standard procedure to "invert" a multivariable function?
I have a function $P(Q,x)$ ($P$ as a function of variables $Q$ and $x$) and I would instead like to known the function $Q(P,x)$ ($Q$ as a function of variables $P$ and $x$).
These functions are in ...
