All Questions
66
questions
0
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0
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28
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Parametric representation of curves
If γ is a $C^1$ curve with parametric representation $φ: [a,b] → R^n$ (i.e. with φ ∈ $C^1$([a, b])) then $L(γ)= \int_a^b ||φ'(t)|| dt$.
I know this hold for $C^1$ but does it also hold for piecewise $...
- 1,913
0
votes
1
answer
171
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Composition of Continuous functions with a finite number of points
Suppose $f$ is continuous everywhere except for a finite number of points and $g$ is continuous everywhere. Then show $g \circ f$ is continuous everywhere except for a finite number of points. Show ...
- 1,913
0
votes
1
answer
36
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proving the existence of a real number c such that the function holds
h: R—>R be a function
h(m+y)=h(m)+h(y)
h(0+0)=h(0)+h(0) –>h(0)=0
h(n)=h(1)+h(1)+...+h(1) (n times)
conclude that h(n)=n*h(1)
Since h(x)xH(1)H(1)< H(1)y
hence H(t)=tH(1)
so ��� c∈R s.t. h(x)=c*x ...
- 553
0
votes
1
answer
192
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How to find all the continuous functions satisfying an equation? [duplicate]
The problem that I want to solve is:
"find all the continuous functions $f\colon \mathbb R\to \mathbb R$ such that for every $x$, $f(f(f(x))) = x $ , I know that f(x) = x is an answer but how can ...
- 540
1
vote
3
answers
114
views
Decreasing positive function less than $x$?
Is there a decreasing function $f$ defined on $(0,∞)$ such that $0<f(x)<x$?
I thought about it and couldn't come up with a conventional function. I was thinking that for all $x>0$ we can find ...
- 3,134
2
votes
1
answer
88
views
Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$
If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that :
$$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$
My attempt
...
user664780
1
vote
3
answers
210
views
Everywhere continuous and differentiable $f : \mathbb{R} → \mathbb{R}$ that is not smooth?
I can't seem to find any counterexamples to the statement "all functions that are continuous and differentiable at every point of the reals are smooth," nor can I find anyone asserting or proving this ...
- 23
8
votes
3
answers
2k
views
Is there a bijective, monotonically increasing, strictly concave function from the reals, to the reals?
I can't come up with a single one.
The range should be the whole of the reals. The best I have is $\log(x)$ but that's only on the positive real line. And there's $f(x) = x$, but this is not strictly ...
- 203
0
votes
1
answer
62
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Proving that a specific function $2^{\mathbb{N}}\to\mathbb{R}$ is injective.
Consider $\mathbb{R}$ as a set of Dedekind cuts. $A \subseteq \mathbb{Q}$ is a Dedekind cut if
$A \notin \{\varnothing,\mathbb{Q}\}$,
$(\forall q \in A)(\forall p \in \mathbb{Q})(p < q \...
- 4,528
5
votes
2
answers
168
views
Existence of Function Taking Every Value Uncountably Many Times
I am stuck on the following problem coming from a plane geometry theorem I'm working on. This is a generalization of that problem which might actually prove easier to handle. All the techniques that ...
- 233
2
votes
2
answers
110
views
Is this theorem true?
If $f(x)+f(y)=f(x+y)$, then:
$f(x)=a x$
where $a$ is a constant.
Is the above statement true? Is there a way of proving it?
The application of this theorem is in the last part of page 52 (second ...
- 1,141
0
votes
0
answers
57
views
Finding functions that intersect at the minimum number of points
Let $f$ and $g$ be (non-constant) functions from $\mathbb{R}^d$ to $\mathbb{R}$.
For a point $x \in \mathbb{R}^d$, let us define the set $S_{f,x} = \{y \in \mathbb{R}^d : f(y) = f(x) \land y \neq x \}$...
- 33
0
votes
1
answer
38
views
Class $C$ functions
How do you prove the following:
In general, a $C^k$ function is contained in $C^{k-1}$ for any $k$.
Why is this true? Thanks for helping.
- 5,164
1
vote
1
answer
81
views
slowest integrable sequence of function
Let $I$ (integrable) be the set of continuous functions $f:\mathbb R_+\to\mathbb R_+$ that are integrable and nonincreasing. Let $D$ (divergent) be the set of continuous functions $g:\mathbb R_+\to\...
- 485
14
votes
2
answers
436
views
Solve $f(x+f(2y))=f(x)+f(y)+y$
Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for each $x$ and $y$ in $\mathbb{R}^+$, $$f(x+f(2y))=f(x)+f(y)+y$$
Note:
$f(x)=x+b$ is a solution for all $b\in\mathbb{R}^+$ but I can not prove ...
