4
votes
Why write codomain instead of image when defining a function.
Here is an easy example.
If $A$ is a set, then $\mathcal P(A)$ is the power set of $A$, i.e. $\{B\mid B\subseteq A\}$. In many contexts, we want to identify $\mathcal P(A)$ with $2^A$, that is the set ...
- 390k
4
votes
How to prove that $f:B\to \mathbb{N}$ by $f(2^k\cdot 3^n)=nk$ with $n,k\in\mathbb{N}$ is a function.
Uniqueness of prime factorization...
- 81
3
votes
There exists a function that satisfies $\sum_{n=1}^\infty |f^{[n]}(x) - f^{[n]}(y)| < \infty$ but is not a contraction?
let
$$
f(x) =\begin{cases} 2x + 1 & 0 \le x < 1 \\
0 & \text{ otherwise}
\end{cases}
$$
That's not a contraction (consider $x = 0, 0.5$ being sent to $1, 2$), but $f^{[2]}$ is everywhere $0$...
- 92.5k
3
votes
An Function that turns $\mathbb Z$ into $\mathbb Q$
Suppose $f:\Bbb Z\to\Bbb Q$ is monotonically increasing. If $f$ is constant, then its range is certainly not the whole of $\Bbb Q$. So suppose $f$ is not constant. Then there exists $n\in\Bbb Z$ such ...
- 63.4k
3
votes
If $f(x)=\mu x-\sin x, x>0$ an is monotonically increasing function. Then find $\mu$
You found correctly that $f'(x) > 0 \iff \mu > \cos x$. If $f$ is monotonically increasing, then $\mu > \cos x$ for all real $x$. This is only true if $\mu > 1$.
It's also true that $$\mu &...
- 8,961
3
votes
Accepted
If $f(x)=\mu x-\sin x, x>0$ an is monotonically increasing function. Then find $\mu$
I think that the problem is that you want
$$\mu-\cos x>0\quad \forall x$$
and in order to ensure that, you need to use the largest possible value of $\cos x$ (rather than the smallest). My ...
- 1,596
2
votes
Why write codomain instead of image when defining a function.
I am not 100% sure if I am going to provide the example you wanted, but let me try.
A couple of days ago, I was reminded of Cantor's diagonal argument. It's a well-known proof of the fact that there ...
- 113
2
votes
Accepted
$|f(x)-f(y)| = 2 |x - y| $, find such functions
You didn't prove that the derivative exists. You're right that $\frac{f(x)-f(c)}{x-c} = \frac{\pm 2|x-c|}{x-c}$ but the sign depends on $x$ and $c$. If the derivative exists then it must be equal to $...
- 8,932
2
votes
Accepted
Prove that Gradient points to Steepest Ascent
By the chain rule
\begin{align} \frac{\Bbb d}{\Bbb dt}\phi_v(t)\vert_{t=0} &= \nabla f(x)\cdot \frac{\Bbb d}{\Bbb dt}(x+tv)\vert_{t=0} \\ &= \nabla f(x)\cdot v \end{align}
So in magnitude we ...
- 2,011
2
votes
Accepted
Continuous and increasing in every variable does not imply continuous?
It does.
For example, consider function on $[0, 1] \times [0, 1]$ that is continuous increasing in both $x$ and $y$, and $f(0, 0) = 0$. We want to show that $\forall \epsilon > 0 \exists \delta >...
- 14.9k
2
votes
Accepted
Mapping from set $A$ surjectively onto itself which is not injective.
I assume you mean, can we have a surjection $A\to A$ that is not injective. Then the answer is yes. By counting reasons, you need that $A$ is infinite, but then it is always possible.
For a simple ...
- 2,561
1
vote
How do you do this question on powersets and surjective functions?
Hint: For each element $b$ of $B$, there is at least one element $x$ of $X$ such that $f(x)=b$, so pick one for each element of $B$ and make a subset of $X$. Can you justify that $f_p$ of that subset ...
- 373k
1
vote
Find $f(2)$ if $f(2 \sin^2(x)+3\sin(x))=1-\tan^2(x)$
Given that:
$\begin{aligned} & f\left(2(\sin x)^2+3 \sin x\right)=1-(\tan x)^2 \\ & 2(\sin x)^2+3 \sin x=2 \\ & \Rightarrow 2(\sin x)^2+3 \sin x-2=0 \\ & \Rightarrow 2(\sin x)^2+4 \sin ...
1
vote
If $f(x)=\mu x-\sin x, x>0$ an is monotonically increasing function. Then find $\mu$
Observe that $f$ is a smooth function, thus for it to be monotonically increasing we require that $f'(x) \ge 0 \ \forall x$, in particular $f'(0) = \mu -1 \ge 0$, which gives the desired result
- 1,322
1
vote
Accepted
Continuity at a point versus continuity after algebraic manipulation
In mathematics, what you've done is known as "function extension." By multiplying your original function by a specific expression, you've created a new function that is continuous and ...
1
vote
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
The idea here is that $\ x_1\ $ and $\ x_3\ $ are fixed but arbitrary real numbers with $\ x_1\le x_3\ .$ Then for any $\ x_2\ge x_3\ $ (no matter how large) you can always express $\ x_3\ $ in the ...
- 27k
1
vote
Accepted
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
To address what your teacher was doing, he is fixing $x_1$ and $x_3$, and then noting that as long as $x_2 \neq x_1$, we can set $\lambda = \frac{x_3 - x_2}{x_1 - x_2}$ to get that $\lambda x_1 + (1 - ...
- 1,643
1
vote
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
Let $f(x)\leqslant B$ for all $x\in\mathbb{R}$, and suppose $f(x_1)<f(x_2)$ for some $x_1,x_2\in\mathbb{R}$. After possibly replacing $f(x)$ with $f(-x)$, we can assume $x_1<x_2$. Thus the slope ...
- 2,193
1
vote
How can I create a function that alternates how much it adds each time?
Try the following function
f(i) = 50*i + 50*FLOOR(i/2)
i
f(i)
0
0
1
50
2
150
3
200
4
300
5
350
6
450
7
500
8
600
..
..
i
$50\,i + 50\,{\rm floor}(i/2)$
- 13.2k
1
vote
Accepted
There exists a function that satisfies $\sum_{n=1}^\infty |f^{[n]}(x) - f^{[n]}(y)| < \infty$ but is not a contraction?
Consider the function
\begin{align*}
f: \mathbb{R} \rightarrow \mathbb{R},
f(x)=\begin{cases} 2^x \pi,& x\in \mathbb{N}, \\
1,& x\in \mathbb{R} \setminus \mathbb{N}. \end{cases}
\end{align*}...
- 17.9k
1
vote
Accepted
Equivalent of sum of arctan(nx)/n²
$f$ is an odd function so that it suffices to investigate its behavior for $x > 0$. One can show that
$$ \tag{*}
- x \ln (x) - \frac x6 \le f(x) \le - x \ln (x) + 5 x
$$
for $0 < x < 1/2$. ...
- 108k
1
vote
Equivalent of sum of arctan(nx)/n²
If, for $x>0$, we use
$$\tan ^{-1}(n x)=\frac \pi 2 -\tan ^{-1}\left(\frac{1}{n x}\right)$$ and
$$\tan ^{-1}\left(\frac{1}{n x}\right)=\sum_{m=0}^\infty\frac {(-1)^m}{(2 m+1)\, x^{2 m+1}\,\, n^{2 m+...
- 256k
1
vote
Accepted
A piecewise function is continuous from a locally compact Hausdorff to metric space.
Clearly $\omega'$ is continuous on $\{f+g > 0\}$. If $x\in \overline{\{f+g > 0\}}$, take a net $x_\alpha \in \{f+g > 0\}$ such that $x_\alpha\to x$. Then from the inequality $|\omega'(x_\...
- 8,932
1
vote
How to define this function so that it is continuous?
I found a definition with floor function that makes the function continuous on entire real line, but this definition is only disguise of a piecewise definition.
$$f(x)=\frac{\arctan(2 \tan (x))+\pi \...
- 1,116
1
vote
How to define this function so that it is continuous?
Maybe better to leave it in its original implicit form, so that the range is continuous.
Converted to its implicit function
$$ \tan (x~y)= 2~\tan(x)$$
and directly plotted on ...
- 39.7k
1
vote
How to define this function so that it is continuous?
The question is about the simplest value for the multi-valued function
$$ y = f(x) := \text{arctan}(2\tan(x))/x. \tag1 $$
A natural place to start is to define the two-variable function
$$ g(x, y) = \...
- 34.3k
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