Are all functions that have a primitive differentiable?
$begingroup$
Are all functions that have a primitive differentiable?
For some background, I know that not all functions that are integrable are differentiable. For example:
$$
f =
begin{cases}
0 & x neq 0 \
1 & x = 0
end{cases}
$$
is integrable over $mathbb{R}$, and $int_{a}^{b} f(x) dx = 0$. However, a function $F(x)$ such that $F'(x) = f(x) space forall x in mathbb{R}$ does not exist.
But I can't find a counterexample to the statement: "All functions that have a primitive are differentiable". These functions are guaranteed to be continuous due to the fundamental theorem of calculus, but not every continuous function is differentiable, hence the question.
Thanks !
Edit: here is a very important comment from @HenningMakholm that I thought would be useful for other students encountering this question
Also, (real) functions that have a primitive are not necessarily continuous. For example,
$$
f(x) =
begin{cases}
0 & text{for } x = 0 \
2x sin(frac{1}{x}) - cos(frac{1}{x}) & text{otherwise}
end{cases}
$$
is discontinuous at $x=0$, but nevertheless the derivative of
$$
F(x) =
begin{cases}
0 & text{ for } x = 0 \
x^2 sin(frac{1}{x}) & text{ otherwise}
end{cases}
$$
Functions that have a primitive do have the intermediate-value property but that is weaker than being continuous.
calculus
$endgroup$
|
show 4 more comments
$begingroup$
Are all functions that have a primitive differentiable?
For some background, I know that not all functions that are integrable are differentiable. For example:
$$
f =
begin{cases}
0 & x neq 0 \
1 & x = 0
end{cases}
$$
is integrable over $mathbb{R}$, and $int_{a}^{b} f(x) dx = 0$. However, a function $F(x)$ such that $F'(x) = f(x) space forall x in mathbb{R}$ does not exist.
But I can't find a counterexample to the statement: "All functions that have a primitive are differentiable". These functions are guaranteed to be continuous due to the fundamental theorem of calculus, but not every continuous function is differentiable, hence the question.
Thanks !
Edit: here is a very important comment from @HenningMakholm that I thought would be useful for other students encountering this question
Also, (real) functions that have a primitive are not necessarily continuous. For example,
$$
f(x) =
begin{cases}
0 & text{for } x = 0 \
2x sin(frac{1}{x}) - cos(frac{1}{x}) & text{otherwise}
end{cases}
$$
is discontinuous at $x=0$, but nevertheless the derivative of
$$
F(x) =
begin{cases}
0 & text{ for } x = 0 \
x^2 sin(frac{1}{x}) & text{ otherwise}
end{cases}
$$
Functions that have a primitive do have the intermediate-value property but that is weaker than being continuous.
calculus
$endgroup$
$begingroup$
Any continuous function on a closed bounded interval has a primitive, but not all continuous functions are differentiable.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:50
$begingroup$
Take a function that is $C^1$ but not $C^2$.
$endgroup$
– dmtri
Dec 29 '18 at 13:51
$begingroup$
@dmtri That is not enough. Your conditions allow for the existence of a discontinuous second derivative.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:52
3
$begingroup$
Note that, famously, this is true for functions defined on an open subset of $mathbb C$.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:50
2
$begingroup$
Also, (real) functions that have a primitive are not necessarily continuous. For example, $$ f(x) = begin{cases} 0 & text{for }x=0 \ 2xsin(1/x) - cos(1/x) & text{otherwise} end{cases} $$ is discontinuous at $x=0$, but is nevertheless the derivative of $$ F(x) = begin{cases} 0 & text{for }x=0 \ x^2sin(1/x) & text{otherwise} end{cases} $$ Functions that have a primitive do have the intermetdiate-value property but that is weaker than being continuous.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:55
|
show 4 more comments
$begingroup$
Are all functions that have a primitive differentiable?
For some background, I know that not all functions that are integrable are differentiable. For example:
$$
f =
begin{cases}
0 & x neq 0 \
1 & x = 0
end{cases}
$$
is integrable over $mathbb{R}$, and $int_{a}^{b} f(x) dx = 0$. However, a function $F(x)$ such that $F'(x) = f(x) space forall x in mathbb{R}$ does not exist.
But I can't find a counterexample to the statement: "All functions that have a primitive are differentiable". These functions are guaranteed to be continuous due to the fundamental theorem of calculus, but not every continuous function is differentiable, hence the question.
Thanks !
Edit: here is a very important comment from @HenningMakholm that I thought would be useful for other students encountering this question
Also, (real) functions that have a primitive are not necessarily continuous. For example,
$$
f(x) =
begin{cases}
0 & text{for } x = 0 \
2x sin(frac{1}{x}) - cos(frac{1}{x}) & text{otherwise}
end{cases}
$$
is discontinuous at $x=0$, but nevertheless the derivative of
$$
F(x) =
begin{cases}
0 & text{ for } x = 0 \
x^2 sin(frac{1}{x}) & text{ otherwise}
end{cases}
$$
Functions that have a primitive do have the intermediate-value property but that is weaker than being continuous.
calculus
$endgroup$
Are all functions that have a primitive differentiable?
For some background, I know that not all functions that are integrable are differentiable. For example:
$$
f =
begin{cases}
0 & x neq 0 \
1 & x = 0
end{cases}
$$
is integrable over $mathbb{R}$, and $int_{a}^{b} f(x) dx = 0$. However, a function $F(x)$ such that $F'(x) = f(x) space forall x in mathbb{R}$ does not exist.
But I can't find a counterexample to the statement: "All functions that have a primitive are differentiable". These functions are guaranteed to be continuous due to the fundamental theorem of calculus, but not every continuous function is differentiable, hence the question.
Thanks !
Edit: here is a very important comment from @HenningMakholm that I thought would be useful for other students encountering this question
Also, (real) functions that have a primitive are not necessarily continuous. For example,
$$
f(x) =
begin{cases}
0 & text{for } x = 0 \
2x sin(frac{1}{x}) - cos(frac{1}{x}) & text{otherwise}
end{cases}
$$
is discontinuous at $x=0$, but nevertheless the derivative of
$$
F(x) =
begin{cases}
0 & text{ for } x = 0 \
x^2 sin(frac{1}{x}) & text{ otherwise}
end{cases}
$$
Functions that have a primitive do have the intermediate-value property but that is weaker than being continuous.
calculus
calculus
edited Dec 30 '18 at 16:08
amWhy
192k28225439
192k28225439
asked Dec 29 '18 at 13:41
ninivertninivert
687
687
$begingroup$
Any continuous function on a closed bounded interval has a primitive, but not all continuous functions are differentiable.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:50
$begingroup$
Take a function that is $C^1$ but not $C^2$.
$endgroup$
– dmtri
Dec 29 '18 at 13:51
$begingroup$
@dmtri That is not enough. Your conditions allow for the existence of a discontinuous second derivative.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:52
3
$begingroup$
Note that, famously, this is true for functions defined on an open subset of $mathbb C$.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:50
2
$begingroup$
Also, (real) functions that have a primitive are not necessarily continuous. For example, $$ f(x) = begin{cases} 0 & text{for }x=0 \ 2xsin(1/x) - cos(1/x) & text{otherwise} end{cases} $$ is discontinuous at $x=0$, but is nevertheless the derivative of $$ F(x) = begin{cases} 0 & text{for }x=0 \ x^2sin(1/x) & text{otherwise} end{cases} $$ Functions that have a primitive do have the intermetdiate-value property but that is weaker than being continuous.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:55
|
show 4 more comments
$begingroup$
Any continuous function on a closed bounded interval has a primitive, but not all continuous functions are differentiable.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:50
$begingroup$
Take a function that is $C^1$ but not $C^2$.
$endgroup$
– dmtri
Dec 29 '18 at 13:51
$begingroup$
@dmtri That is not enough. Your conditions allow for the existence of a discontinuous second derivative.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:52
3
$begingroup$
Note that, famously, this is true for functions defined on an open subset of $mathbb C$.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:50
2
$begingroup$
Also, (real) functions that have a primitive are not necessarily continuous. For example, $$ f(x) = begin{cases} 0 & text{for }x=0 \ 2xsin(1/x) - cos(1/x) & text{otherwise} end{cases} $$ is discontinuous at $x=0$, but is nevertheless the derivative of $$ F(x) = begin{cases} 0 & text{for }x=0 \ x^2sin(1/x) & text{otherwise} end{cases} $$ Functions that have a primitive do have the intermetdiate-value property but that is weaker than being continuous.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:55
$begingroup$
Any continuous function on a closed bounded interval has a primitive, but not all continuous functions are differentiable.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:50
$begingroup$
Any continuous function on a closed bounded interval has a primitive, but not all continuous functions are differentiable.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:50
$begingroup$
Take a function that is $C^1$ but not $C^2$.
$endgroup$
– dmtri
Dec 29 '18 at 13:51
$begingroup$
Take a function that is $C^1$ but not $C^2$.
$endgroup$
– dmtri
Dec 29 '18 at 13:51
$begingroup$
@dmtri That is not enough. Your conditions allow for the existence of a discontinuous second derivative.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:52
$begingroup$
@dmtri That is not enough. Your conditions allow for the existence of a discontinuous second derivative.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:52
3
3
$begingroup$
Note that, famously, this is true for functions defined on an open subset of $mathbb C$.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:50
$begingroup$
Note that, famously, this is true for functions defined on an open subset of $mathbb C$.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:50
2
2
$begingroup$
Also, (real) functions that have a primitive are not necessarily continuous. For example, $$ f(x) = begin{cases} 0 & text{for }x=0 \ 2xsin(1/x) - cos(1/x) & text{otherwise} end{cases} $$ is discontinuous at $x=0$, but is nevertheless the derivative of $$ F(x) = begin{cases} 0 & text{for }x=0 \ x^2sin(1/x) & text{otherwise} end{cases} $$ Functions that have a primitive do have the intermetdiate-value property but that is weaker than being continuous.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:55
$begingroup$
Also, (real) functions that have a primitive are not necessarily continuous. For example, $$ f(x) = begin{cases} 0 & text{for }x=0 \ 2xsin(1/x) - cos(1/x) & text{otherwise} end{cases} $$ is discontinuous at $x=0$, but is nevertheless the derivative of $$ F(x) = begin{cases} 0 & text{for }x=0 \ x^2sin(1/x) & text{otherwise} end{cases} $$ Functions that have a primitive do have the intermetdiate-value property but that is weaker than being continuous.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:55
|
show 4 more comments
3 Answers
3
active
oldest
votes
$begingroup$
A primitive for $f(x)=x^{1/3}$ is $F(x)=frac{3}{4}x^{4/3}$, but $f'(0)$ doesn't exist, because $$limlimits_{xto0}dfrac{f(x)-f(0)}{x-0}=limlimits_{xto0}dfrac{x^{1/3}}{x}=limlimits_{xto0}dfrac{1}{x^{2/3}}=infty$$
$endgroup$
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
2
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
add a comment |
$begingroup$
Even more simply: $f(x) := |x|$.
$endgroup$
3
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
add a comment |
$begingroup$
Any bounded function that has discontinuity at a single point is integrable but of course that function will be non differentiable.
$endgroup$
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
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%2f3055852%2fare-all-functions-that-have-a-primitive-differentiable%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
$begingroup$
A primitive for $f(x)=x^{1/3}$ is $F(x)=frac{3}{4}x^{4/3}$, but $f'(0)$ doesn't exist, because $$limlimits_{xto0}dfrac{f(x)-f(0)}{x-0}=limlimits_{xto0}dfrac{x^{1/3}}{x}=limlimits_{xto0}dfrac{1}{x^{2/3}}=infty$$
$endgroup$
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
2
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
add a comment |
$begingroup$
A primitive for $f(x)=x^{1/3}$ is $F(x)=frac{3}{4}x^{4/3}$, but $f'(0)$ doesn't exist, because $$limlimits_{xto0}dfrac{f(x)-f(0)}{x-0}=limlimits_{xto0}dfrac{x^{1/3}}{x}=limlimits_{xto0}dfrac{1}{x^{2/3}}=infty$$
$endgroup$
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
2
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
add a comment |
$begingroup$
A primitive for $f(x)=x^{1/3}$ is $F(x)=frac{3}{4}x^{4/3}$, but $f'(0)$ doesn't exist, because $$limlimits_{xto0}dfrac{f(x)-f(0)}{x-0}=limlimits_{xto0}dfrac{x^{1/3}}{x}=limlimits_{xto0}dfrac{1}{x^{2/3}}=infty$$
$endgroup$
A primitive for $f(x)=x^{1/3}$ is $F(x)=frac{3}{4}x^{4/3}$, but $f'(0)$ doesn't exist, because $$limlimits_{xto0}dfrac{f(x)-f(0)}{x-0}=limlimits_{xto0}dfrac{x^{1/3}}{x}=limlimits_{xto0}dfrac{1}{x^{2/3}}=infty$$
edited Dec 29 '18 at 14:01
answered Dec 29 '18 at 13:51
Martín Vacas VignoloMartín Vacas Vignolo
3,810623
3,810623
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
2
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
add a comment |
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
2
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
$begingroup$
Almost. What you wrote is $f'(x)$ for $xne0$; you need an additional argument to ensure that $f'(0)$ does not exist.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:55
2
2
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
$begingroup$
@AndrésE.Caicedo I edited to be more clarifying
$endgroup$
– Martín Vacas Vignolo
Dec 29 '18 at 14:02
add a comment |
$begingroup$
Even more simply: $f(x) := |x|$.
$endgroup$
3
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
add a comment |
$begingroup$
Even more simply: $f(x) := |x|$.
$endgroup$
3
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
add a comment |
$begingroup$
Even more simply: $f(x) := |x|$.
$endgroup$
Even more simply: $f(x) := |x|$.
answered Dec 29 '18 at 14:55
The_SympathizerThe_Sympathizer
7,4852245
7,4852245
3
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
add a comment |
3
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
3
3
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
$begingroup$
Of course, how did I not think about that ? I already accepted the @Martin's answer, but thanks for the example ! For further reference, I'll just state here that $int |x| dx = frac{1}{2}x|x|$
$endgroup$
– ninivert
Dec 29 '18 at 15:26
add a comment |
$begingroup$
Any bounded function that has discontinuity at a single point is integrable but of course that function will be non differentiable.
$endgroup$
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
add a comment |
$begingroup$
Any bounded function that has discontinuity at a single point is integrable but of course that function will be non differentiable.
$endgroup$
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
add a comment |
$begingroup$
Any bounded function that has discontinuity at a single point is integrable but of course that function will be non differentiable.
$endgroup$
Any bounded function that has discontinuity at a single point is integrable but of course that function will be non differentiable.
answered Dec 29 '18 at 13:53
user630002
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
add a comment |
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
$begingroup$
@Martin has justified my statement.
$endgroup$
– user630002
Dec 29 '18 at 13:54
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%2f3055852%2fare-all-functions-that-have-a-primitive-differentiable%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
$begingroup$
Any continuous function on a closed bounded interval has a primitive, but not all continuous functions are differentiable.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:50
$begingroup$
Take a function that is $C^1$ but not $C^2$.
$endgroup$
– dmtri
Dec 29 '18 at 13:51
$begingroup$
@dmtri That is not enough. Your conditions allow for the existence of a discontinuous second derivative.
$endgroup$
– Andrés E. Caicedo
Dec 29 '18 at 13:52
3
$begingroup$
Note that, famously, this is true for functions defined on an open subset of $mathbb C$.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:50
2
$begingroup$
Also, (real) functions that have a primitive are not necessarily continuous. For example, $$ f(x) = begin{cases} 0 & text{for }x=0 \ 2xsin(1/x) - cos(1/x) & text{otherwise} end{cases} $$ is discontinuous at $x=0$, but is nevertheless the derivative of $$ F(x) = begin{cases} 0 & text{for }x=0 \ x^2sin(1/x) & text{otherwise} end{cases} $$ Functions that have a primitive do have the intermetdiate-value property but that is weaker than being continuous.
$endgroup$
– Henning Makholm
Dec 29 '18 at 15:55