All Questions
66
questions
2
votes
1
answer
171
views
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 $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 317
0
votes
3
answers
72
views
{$\frac{1}{x}$} , $0<x\leq1$ in terms of {$x$}, $x\geq 1$, {$x$} is fractional part of $x$
$f(x)$ $=$ {$\frac{1}{x}$}, $0<x\leq1$ where {$x$} denotes the fractional part of $x$.
$g(x) =$ {$x$}, $x\geq 1$
I want an expression for $f(x)$ in terms of x and $g(x)$.
My try-
If $x\in \mathbb{Z}...
user826419
3
votes
1
answer
81
views
Suppose $\sum_{n\ge 1} |a_n| = A<\infty.$ Under what conditions is $\sum_{n\ge 1} \epsilon_n a_n = [-A,A]$, for $\epsilon_n \in \{-1,1\}$?
Consider the space of sequences:
$$
\mathcal{E} = \{\{\epsilon_n\}_{n= 1}^{\infty}: \epsilon_n = \pm 1\}
$$
This can be considered a "random choice of sign" in the probabilistic context, for ...
- 8,369
0
votes
3
answers
94
views
Is it true that there is a bijection $[0, 1) \to \mathbb{R}$?
Is there is a bijection from $[0,1)$ to $\mathbb{R}$?
I thought of an instance, $$\frac{\sqrt{x(1-x)}}{x-1}.$$
- 127
0
votes
1
answer
42
views
Explain a confusing bound for the integral of a decreasing function.
I am reading a solution of an exercise. In the solution, it says the following:
Consider $g(x,t):=\frac{x}{(1+tx^{2})t^{\alpha}}$, where $x\in (0,\infty)$, $t=1,2,3,\cdots$ and $\alpha>\frac{1}{2}$...
- 5,930
-2
votes
2
answers
50
views
Let $a$ be a real number such that $a > 0$. Show that the function $f : [a, +\infty) \to \mathbb R, f(x) = \tfrac{1}{x}$ is uniformly continuous.
Let $a$ be a real number such that $a > 0$. Show that the function $$f : [a, +\infty) \to \mathbb R, f(x) = \dfrac{1}{x}$$
is uniformly continuous.
- 1
0
votes
1
answer
106
views
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have $|f(x) − f(y)| < |x − y|$
Let $a$ and $b$ two real numbers such that $a < b$ and $f : [a, b] \to [a, b]$.
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have
$|f(x) − f(y)| < |x − y|$. Show that ...
- 27
3
votes
2
answers
86
views
If following actions allowed, Find $F(2002,2020,2200)?$
If following actions allowed,Find $F(2002,2020,2200)?$
$$ F(x+t,y+t,z+t)=t+F(x,y,z);$$
$$ F(xt,yt,zt)=tF(x,y,z);$$
$$ F(x,y,z)=F(y,x,z)=F(x,z,y)$$
where x,y,z,t are real numbers.
My attempt:
$F(0,0,0)...
- 59
0
votes
2
answers
89
views
Is there a name for a real-valued function whose input is also real?
I'm trying to write a sentence about a function $f:\mathbb{R}\to \mathbb{R}$, and I want to refer to it as real valued or as a scalar function, or some similar term, but I want that term to also ...
- 1,361
9
votes
3
answers
153
views
How different can $f(g(x))$ and $g(f(x))$ be?
Given $f,g: \mathbb{R} \rightarrow \mathbb{R}$, how "different" can $f(g(x))$ and $g(f(x))$ be?
By "how different" I mean:
Given two real-valued functions $a,b$ do there exist two real-valued ...
- 2,471
-1
votes
1
answer
132
views
Is a function $f(x)=\ln({x^2-1})$ even and symmetric
We have a function:
$$
f(x)=\ln(x^2-1)
$$
The function is symmetric because: $D_f=(-\infty,-1) \ \cup\ (1,\infty)$
I understand this as if we would multipy this by $-1$ we would get the same $D_f$
...
- 2,527
2
votes
2
answers
208
views
Why is this function continuous on $\mathbb R$?
Let $f:\Bbb{R}\to\mathbb R$ be a function with $f(0) = 1$ and $f(x+y) \le f(x)f(y)$ for all $x, y \in \Bbb{R}$. Prove that if $f$ is continuous at $0$, then $f$ is continuous on $\Bbb{R}$?
THOUGHTS:
...
- 160
1
vote
2
answers
231
views
Set of constant functions are uncountable.
Let $F=$ $\{$ $f: [0,1] \rightarrow \mathbb{R}$ $:$ $f$ is constant$ \} $. I must show that $F$ is uncountable.
Note, that for any $f \in F$, and any $c\in \mathbb{R}$, I will denote the constant ...
user643073
2
votes
0
answers
72
views
Proving integral of a continuous function is continuous
Let $U \subset \mathbb{R}^N$ be an open set, let $f : U \times [a, b] \to \mathbb{R}$ be a continuous function. Consider the function $$g(x):= \int_a^b f(x,y) \,dy$$
with $x \in U$.
i) Prove ...
- 1,913
2
votes
2
answers
461
views
Strictly increasing bounded function of class $C^1$
Let $f:\mathbb{R} \to \mathbb{R}$ be a strictly increasing bounded function of class $C^1$. Prove that there exists a sequence $\{x_n\}_n$ of real numbers such that $x_n\to\infty$ and $\lim_{n \to \...
- 1,913