Is there an injective, monotonically increasing, strictly concave function from the reals, to the reals?Example of continuous but not absolutely continuous strictly increasing functionCan you build metric space theory without the real numbers?Proof by induction: prove that if $x_0>3$ then the following sequence is strictly increasing…A continuous function $f:Bbb Rto Bbb R$ is injective if and only if it is strictly increasing or strictly decreasingIs there a “jagged” real-valued function that is “smooth” in cardinalities greater than the reals?examples of first strictly concave then convex function?Inverse of any strictly monotonic increasing function defined over a fixed domain and range.Continuity of $argmax$ of a strictly concave functionstrictly increasing function from reals to reals which is never an algebraic numberAt which value (over $mathbbR^+$) is the gamma function strictly increasing?
Multi tool use
What if you are holding an Iron Flask with a demon inside and walk into Antimagic Field?
Bridge building with irregular planks
How to cover method return statement in Apex Class?
Does an advisor owe his/her student anything? Will an advisor keep a PhD student only out of pity?
What features enable the Su-25 Frogfoot to operate with such a wide variety of fuels?
Why did the EU agree to delay the Brexit deadline?
What is the highest possible scrabble score for placing a single tile
Unexpected behavior of the procedure `Area` on the object 'Polygon'
Is there an injective, monotonically increasing, strictly concave function from the reals, to the reals?
What are the advantages of simplicial model categories over non-simplicial ones?
What to do when eye contact makes your subordinate uncomfortable?
Mixing PEX brands
Fear of getting stuck on one programming language / technology that is not used in my country
What does "Scientists rise up against statistical significance" mean? (Comment in Nature)
How much character growth crosses the line into breaking the character
Why is it that I can sometimes guess the next note?
Does the Linux kernel need a file system to run?
Why is short-wave infrared portion of electromagnetic spectrum so sensitive to fire?
Can a college of swords bard use blade flourish on an OA from dissonant whispers?
Why does the Sun have different day lengths, but not the gas giants?
This is why we puzzle
System.QueryException unexpected token
What is going on with 'gets(stdin)' on the site coderbyte?
Is there a RAID 0 Equivalent for RAM?
Is there an injective, monotonically increasing, strictly concave function from the reals, to the reals?
Example of continuous but not absolutely continuous strictly increasing functionCan you build metric space theory without the real numbers?Proof by induction: prove that if $x_0>3$ then the following sequence is strictly increasing…A continuous function $f:Bbb Rto Bbb R$ is injective if and only if it is strictly increasing or strictly decreasingIs there a “jagged” real-valued function that is “smooth” in cardinalities greater than the reals?examples of first strictly concave then convex function?Inverse of any strictly monotonic increasing function defined over a fixed domain and range.Continuity of $argmax$ of a strictly concave functionstrictly increasing function from reals to reals which is never an algebraic numberAt which value (over $mathbbR^+$) is the gamma function strictly increasing?
$begingroup$
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 concave. And $-e^-x$ only maps to half of the real line.
Any ideas?
real-analysis functions recreational-mathematics real-numbers
asked 3 hours ago
cammilcammil
1214
$endgroup$
add a comment |
$begingroup$
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 concave. And $-e^-x$ only maps to half of the real line.
Any ideas?
real-analysis functions recreational-mathematics real-numbers
asked 3 hours ago
cammilcammil
1214
$endgroup$
5
$begingroup$
$f(x) = -e^-x$?
$endgroup$
– Daniel Schepler
3 hours ago
1
$begingroup$
@DanielSchepler I was just about to write the same, +1.
$endgroup$
– Michael Hoppe
3 hours ago
$begingroup$
Sorry, I should have made clear, it should map to the whole of the reals. (What's the mathematical term for that?)
$endgroup$
– cammil
3 hours ago
1
$begingroup$
@cammil a surjection (i.e. a function whose range is equal to its codomain).
$endgroup$
– Jake
3 hours ago
$begingroup$
If you start with the lower right branch of the hyperbola $xy=-1$ and transform the coordinates to slope the $x$ axis upward to the right and the $y$ axis rightward toward the top, you will have another choice.
$endgroup$
– Ross Millikan
1 hour ago
add a comment |
$begingroup$
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 concave. And $-e^-x$ only maps to half of the real line.
Any ideas?
real-analysis functions recreational-mathematics real-numbers
asked 3 hours ago
cammilcammil
1214
$endgroup$
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 concave. And $-e^-x$ only maps to half of the real line.
Any ideas?
real-analysis functions recreational-mathematics real-numbers
real-analysis functions recreational-mathematics real-numbers
asked 3 hours ago
cammilcammil
1214
asked 3 hours ago
cammilcammil
1214
edited 3 hours ago
cammil
asked 3 hours ago
cammilcammil
1214
asked 3 hours ago
cammilcammil
1214
asked 3 hours ago
cammilcammil
1214
1214
5
$begingroup$
$f(x) = -e^-x$?
$endgroup$
– Daniel Schepler
3 hours ago
1
$begingroup$
@DanielSchepler I was just about to write the same, +1.
$endgroup$
– Michael Hoppe
3 hours ago
$begingroup$
Sorry, I should have made clear, it should map to the whole of the reals. (What's the mathematical term for that?)
$endgroup$
– cammil
3 hours ago
1
$begingroup$
@cammil a surjection (i.e. a function whose range is equal to its codomain).
$endgroup$
– Jake
3 hours ago
$begingroup$
If you start with the lower right branch of the hyperbola $xy=-1$ and transform the coordinates to slope the $x$ axis upward to the right and the $y$ axis rightward toward the top, you will have another choice.
$endgroup$
– Ross Millikan
1 hour ago
add a comment |
5
$begingroup$
$f(x) = -e^-x$?
$endgroup$
– Daniel Schepler
3 hours ago
1
$begingroup$
@DanielSchepler I was just about to write the same, +1.
$endgroup$
– Michael Hoppe
3 hours ago
$begingroup$
Sorry, I should have made clear, it should map to the whole of the reals. (What's the mathematical term for that?)
$endgroup$
– cammil
3 hours ago
1
$begingroup$
@cammil a surjection (i.e. a function whose range is equal to its codomain).
$endgroup$
– Jake
3 hours ago
$begingroup$
If you start with the lower right branch of the hyperbola $xy=-1$ and transform the coordinates to slope the $x$ axis upward to the right and the $y$ axis rightward toward the top, you will have another choice.
$endgroup$
– Ross Millikan
1 hour ago
5
5
$begingroup$
$f(x) = -e^-x$?
$endgroup$
– Daniel Schepler
3 hours ago
$begingroup$
$f(x) = -e^-x$?
$endgroup$
– Daniel Schepler
3 hours ago
1
1
$begingroup$
@DanielSchepler I was just about to write the same, +1.
$endgroup$
– Michael Hoppe
3 hours ago
$begingroup$
@DanielSchepler I was just about to write the same, +1.
$endgroup$
– Michael Hoppe
3 hours ago
$begingroup$
Sorry, I should have made clear, it should map to the whole of the reals. (What's the mathematical term for that?)
$endgroup$
– cammil
3 hours ago
$begingroup$
Sorry, I should have made clear, it should map to the whole of the reals. (What's the mathematical term for that?)
$endgroup$
– cammil
3 hours ago
1
1
$begingroup$
@cammil a surjection (i.e. a function whose range is equal to its codomain).
$endgroup$
– Jake
3 hours ago
$begingroup$
@cammil a surjection (i.e. a function whose range is equal to its codomain).
$endgroup$
– Jake
3 hours ago
$begingroup$
If you start with the lower right branch of the hyperbola $xy=-1$ and transform the coordinates to slope the $x$ axis upward to the right and the $y$ axis rightward toward the top, you will have another choice.
$endgroup$
– Ross Millikan
1 hour ago
$begingroup$
If you start with the lower right branch of the hyperbola $xy=-1$ and transform the coordinates to slope the $x$ axis upward to the right and the $y$ axis rightward toward the top, you will have another choice.
$endgroup$
– Ross Millikan
1 hour ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
$$
f(x) = x-e^-x
$$
is such a function. Since $f''(x) = -e^-x$ is always negative, it is strictly concave, and it's not hard to show it hits every real.
Even better,
$$
f(x) = 2x -sqrt1+3x^2
$$
has $f''(x) = -3(1+3x^2)^-3/2 < 0$ everywhere and the explicit inverse $f^-1(x) = 2x+sqrt1+3x^2$, clearly defined for all $x$.
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
$endgroup$
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
add a comment |
$begingroup$
How about
$f(x)=left{beginarraycc ln(x+1)& &xge 0\1-e^-x& &x<0endarrayright.$
answered 3 hours ago
paw88789paw88789
29.4k12349
$endgroup$
add a comment |
$begingroup$
$F(x) = pi x+ int_0^x arctan (-t),dt$ is an example. Many more examples like this one can be constructed.
answered 2 hours ago
zhw.zhw.
74.5k43175
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3158452%2fis-there-an-injective-monotonically-increasing-strictly-concave-function-from%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$
f(x) = x-e^-x
$$
is such a function. Since $f''(x) = -e^-x$ is always negative, it is strictly concave, and it's not hard to show it hits every real.
Even better,
$$
f(x) = 2x -sqrt1+3x^2
$$
has $f''(x) = -3(1+3x^2)^-3/2 < 0$ everywhere and the explicit inverse $f^-1(x) = 2x+sqrt1+3x^2$, clearly defined for all $x$.
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
$endgroup$
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
add a comment |
$begingroup$
$$
f(x) = x-e^-x
$$
is such a function. Since $f''(x) = -e^-x$ is always negative, it is strictly concave, and it's not hard to show it hits every real.
Even better,
$$
f(x) = 2x -sqrt1+3x^2
$$
has $f''(x) = -3(1+3x^2)^-3/2 < 0$ everywhere and the explicit inverse $f^-1(x) = 2x+sqrt1+3x^2$, clearly defined for all $x$.
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
$endgroup$
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
add a comment |
$begingroup$
$$
f(x) = x-e^-x
$$
is such a function. Since $f''(x) = -e^-x$ is always negative, it is strictly concave, and it's not hard to show it hits every real.
Even better,
$$
f(x) = 2x -sqrt1+3x^2
$$
has $f''(x) = -3(1+3x^2)^-3/2 < 0$ everywhere and the explicit inverse $f^-1(x) = 2x+sqrt1+3x^2$, clearly defined for all $x$.
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
$endgroup$
$$
f(x) = x-e^-x
$$
is such a function. Since $f''(x) = -e^-x$ is always negative, it is strictly concave, and it's not hard to show it hits every real.
Even better,
$$
f(x) = 2x -sqrt1+3x^2
$$
has $f''(x) = -3(1+3x^2)^-3/2 < 0$ everywhere and the explicit inverse $f^-1(x) = 2x+sqrt1+3x^2$, clearly defined for all $x$.
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
edited 2 hours ago
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
answered 3 hours ago
eyeballfrogeyeballfrog
6,664630
6,664630
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
add a comment |
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
$begingroup$
+1 (All hail the Hypnotoad!) Dare I ask how you found the second example? I had to work a bit even to check the inverse formula. I assume I'm missing something really neat.
$endgroup$
– Calum Gilhooley
2 hours ago
add a comment |
$begingroup$
How about
$f(x)=left{beginarraycc ln(x+1)& &xge 0\1-e^-x& &x<0endarrayright.$
answered 3 hours ago
paw88789paw88789
29.4k12349
$endgroup$
add a comment |
$begingroup$
How about
$f(x)=left{beginarraycc ln(x+1)& &xge 0\1-e^-x& &x<0endarrayright.$
answered 3 hours ago
paw88789paw88789
29.4k12349
$endgroup$
add a comment |
$begingroup$
How about
$f(x)=left{beginarraycc ln(x+1)& &xge 0\1-e^-x& &x<0endarrayright.$
answered 3 hours ago
paw88789paw88789
29.4k12349
$endgroup$
How about
$f(x)=left{beginarraycc ln(x+1)& &xge 0\1-e^-x& &x<0endarrayright.$
answered 3 hours ago
paw88789paw88789
29.4k12349
answered 3 hours ago
paw88789paw88789
29.4k12349
answered 3 hours ago
paw88789paw88789
29.4k12349
answered 3 hours ago
paw88789paw88789
29.4k12349
29.4k12349
add a comment |
add a comment |
$begingroup$
$F(x) = pi x+ int_0^x arctan (-t),dt$ is an example. Many more examples like this one can be constructed.
answered 2 hours ago
zhw.zhw.
74.5k43175
$endgroup$
add a comment |
$begingroup$
$F(x) = pi x+ int_0^x arctan (-t),dt$ is an example. Many more examples like this one can be constructed.
answered 2 hours ago
zhw.zhw.
74.5k43175
$endgroup$
add a comment |
$begingroup$
$F(x) = pi x+ int_0^x arctan (-t),dt$ is an example. Many more examples like this one can be constructed.
answered 2 hours ago
zhw.zhw.
74.5k43175
$endgroup$
$F(x) = pi x+ int_0^x arctan (-t),dt$ is an example. Many more examples like this one can be constructed.
answered 2 hours ago
zhw.zhw.
74.5k43175
answered 2 hours ago
zhw.zhw.
74.5k43175
answered 2 hours ago
zhw.zhw.
74.5k43175
answered 2 hours ago
zhw.zhw.
74.5k43175
74.5k43175
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3158452%2fis-there-an-injective-monotonically-increasing-strictly-concave-function-from%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
FRbOa,4f9ke 74O
5
$begingroup$
$f(x) = -e^-x$?
$endgroup$
– Daniel Schepler
3 hours ago
1
$begingroup$
@DanielSchepler I was just about to write the same, +1.
$endgroup$
– Michael Hoppe
3 hours ago
$begingroup$
Sorry, I should have made clear, it should map to the whole of the reals. (What's the mathematical term for that?)
$endgroup$
– cammil
3 hours ago
1
$begingroup$
@cammil a surjection (i.e. a function whose range is equal to its codomain).
$endgroup$
– Jake
3 hours ago
$begingroup$
If you start with the lower right branch of the hyperbola $xy=-1$ and transform the coordinates to slope the $x$ axis upward to the right and the $y$ axis rightward toward the top, you will have another choice.
$endgroup$
– Ross Millikan
1 hour ago