All Questions
10
questions
0
votes
1
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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
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
- 33
0
votes
4
answers
67
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Rudin PMA 4.31 - does the elements of $E$ have to be ordered (smallest to the biggest)?
Here is a reformulation of Rudin PMA $4.31$ remark:
Let $a$ and $b$ be two real numbers such that $a < b$, let $E$ be any countable subset of the open interval $(a,b)$, and let the elements of $E$ ...
- 1,231
0
votes
2
answers
74
views
Nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = (f(x))^2$
Are there any nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the identity $f(f(x)) = (f(x))^2$ $\forall x \in \mathbb{R}$?
1
vote
0
answers
17
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Looking for a sigmoid-like function with different properties
I am looking for a function that is 0 at 0, 1 at 1, increases to a predetermined $x_1$ relatively quickly, acquires a derivative close to 0 (but doesn't actually plateau) then starts increasing at $...
- 3,439
2
votes
1
answer
169
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Proving if a function is continuous and not one-one then it has many such points.
Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 317
2
votes
2
answers
53
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what are all functions with $x>1$ and $y>1$ $\rho$ that follows $\rho(xy)=\frac{1}{\frac{y}{\rho(x)}+\frac{x}{\rho(y)}}$ and is continuous
What are all functions with $x>1$ and $y>1$ $\rho$ that follows $$\rho(xy)=\frac{1}{\frac{y}{\rho(x)}+\frac{x}{\rho(y)}}$$ and is continuous
If this doesn't have any solutions then prove no such ...
user872441
0
votes
1
answer
135
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true or false- continuous functions
I'm having some hard time deciding if those sentences are true or false:
$1$. If $f$ is continuous on $\mathbb{R}$ then if $\left|f(x)-x\right|<1$ for every $x$ on $\mathbb{R}$ then $f$ is getting ...
- 65
5
votes
2
answers
90
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Finding the number of continuous functions
Question:
Find the number of continuous function(s) $f:[0, 1]\to\mathbb{R}$ satisfying $$\int_0^1f(x)\text{d}x=\frac{1}{3}+\int_0^1f^2(x^2)\text{d}x$$
My approach:
I put $x^2=t$, giving $2x\text{d}x=...
- 127
2
votes
2
answers
207
views
Why is this function continuous on $\mathbb R$?
Let $f:\Bbb{R}\to\mathbb R$ be a function with $f(0) = 1$ and $f(x+y) \le f(x)f(y)$ for all $x, y \in \Bbb{R}$. Prove that if $f$ is continuous at $0$, then $f$ is continuous on $\Bbb{R}$?
THOUGHTS:
...
- 160
2
votes
0
answers
72
views
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 ...
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