Is $F$ continuously differentiable at $x=0$? [duplicate]
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This question already has an answer here:
Differentiable but not continuously differentiable.
1 answer
a) Find all the values of $alpha$ such that $f'(0)$ exists. b) Find all the values of $alpha$ such that $f$ is of bounded variation on $[0,1]$
1 answer
Do you have any tips? Especially for the second part? IS it enough to say since $f(x)$ is continuous at $x=0$ and $f$ is differentiable at $x=0$. Is $f$ continuously differentiable? Here is the question.
Let
$$
f(x)=
begin{cases}
x^2sinbig(frac{1}{x}big) & text{ if }xneq 0\
0 & text{ if }x= 0
end{cases}
$$
Show that $f$ is differentiable at $x=0$ and compute $f^prime(0)$. Is $F$ continuously differentiable at $x=0$?
Edit: For the second part, I used the fundamental theorem of calculus part 2. f is continuous and according to that theorem, so F is continuously differentiable.
P.S: I don't think this question is a duplicate of another question. In that question, it is asking for the derivative of f. But, in this question, it is asking for the integral of f as the capital F is a symbol for the integral of f. I don't understand why you keep insisting that this question is a duplicate.
real-analysis analysis
asked Nov 22 at 18:44
bebe
146
marked as duplicate by Lord Shark the Unknown, Brahadeesh, user10354138, Martin Sleziak, onurcanbektas Nov 23 at 9:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
up vote
0
down vote
favorite
This question already has an answer here:
Differentiable but not continuously differentiable.
1 answer
a) Find all the values of $alpha$ such that $f'(0)$ exists. b) Find all the values of $alpha$ such that $f$ is of bounded variation on $[0,1]$
1 answer
Do you have any tips? Especially for the second part? IS it enough to say since $f(x)$ is continuous at $x=0$ and $f$ is differentiable at $x=0$. Is $f$ continuously differentiable? Here is the question.
Let
$$
f(x)=
begin{cases}
x^2sinbig(frac{1}{x}big) & text{ if }xneq 0\
0 & text{ if }x= 0
end{cases}
$$
Show that $f$ is differentiable at $x=0$ and compute $f^prime(0)$. Is $F$ continuously differentiable at $x=0$?
Edit: For the second part, I used the fundamental theorem of calculus part 2. f is continuous and according to that theorem, so F is continuously differentiable.
P.S: I don't think this question is a duplicate of another question. In that question, it is asking for the derivative of f. But, in this question, it is asking for the integral of f as the capital F is a symbol for the integral of f. I don't understand why you keep insisting that this question is a duplicate.
real-analysis analysis
asked Nov 22 at 18:44
bebe
146
marked as duplicate by Lord Shark the Unknown, Brahadeesh, user10354138, Martin Sleziak, onurcanbektas Nov 23 at 9:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.)
– TonyK
Nov 22 at 18:50
No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me
– bebe
Nov 22 at 19:06
@TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question
– bebe
Nov 22 at 19:09
That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran
– bebe
Nov 22 at 19:12
This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...)
– TonyK
Nov 22 at 19:15
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This question already has an answer here:
Differentiable but not continuously differentiable.
1 answer
a) Find all the values of $alpha$ such that $f'(0)$ exists. b) Find all the values of $alpha$ such that $f$ is of bounded variation on $[0,1]$
1 answer
Do you have any tips? Especially for the second part? IS it enough to say since $f(x)$ is continuous at $x=0$ and $f$ is differentiable at $x=0$. Is $f$ continuously differentiable? Here is the question.
Let
$$
f(x)=
begin{cases}
x^2sinbig(frac{1}{x}big) & text{ if }xneq 0\
0 & text{ if }x= 0
end{cases}
$$
Show that $f$ is differentiable at $x=0$ and compute $f^prime(0)$. Is $F$ continuously differentiable at $x=0$?
Edit: For the second part, I used the fundamental theorem of calculus part 2. f is continuous and according to that theorem, so F is continuously differentiable.
P.S: I don't think this question is a duplicate of another question. In that question, it is asking for the derivative of f. But, in this question, it is asking for the integral of f as the capital F is a symbol for the integral of f. I don't understand why you keep insisting that this question is a duplicate.
real-analysis analysis
asked Nov 22 at 18:44
bebe
146
This question already has an answer here:
Differentiable but not continuously differentiable.
1 answer
a) Find all the values of $alpha$ such that $f'(0)$ exists. b) Find all the values of $alpha$ such that $f$ is of bounded variation on $[0,1]$
1 answer
Do you have any tips? Especially for the second part? IS it enough to say since $f(x)$ is continuous at $x=0$ and $f$ is differentiable at $x=0$. Is $f$ continuously differentiable? Here is the question.
Let
$$
f(x)=
begin{cases}
x^2sinbig(frac{1}{x}big) & text{ if }xneq 0\
0 & text{ if }x= 0
end{cases}
$$
Show that $f$ is differentiable at $x=0$ and compute $f^prime(0)$. Is $F$ continuously differentiable at $x=0$?
Edit: For the second part, I used the fundamental theorem of calculus part 2. f is continuous and according to that theorem, so F is continuously differentiable.
P.S: I don't think this question is a duplicate of another question. In that question, it is asking for the derivative of f. But, in this question, it is asking for the integral of f as the capital F is a symbol for the integral of f. I don't understand why you keep insisting that this question is a duplicate.
This question already has an answer here:
Differentiable but not continuously differentiable.
1 answer
a) Find all the values of $alpha$ such that $f'(0)$ exists. b) Find all the values of $alpha$ such that $f$ is of bounded variation on $[0,1]$
1 answer
real-analysis analysis
real-analysis analysis
asked Nov 22 at 18:44
bebe
146
asked Nov 22 at 18:44
bebe
146
edited Nov 23 at 14:00
asked Nov 22 at 18:44
bebe
146
asked Nov 22 at 18:44
bebe
146
asked Nov 22 at 18:44
bebe
146
146
marked as duplicate by Lord Shark the Unknown, Brahadeesh, user10354138, Martin Sleziak, onurcanbektas Nov 23 at 9:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Lord Shark the Unknown, Brahadeesh, user10354138, Martin Sleziak, onurcanbektas Nov 23 at 9:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.)
– TonyK
Nov 22 at 18:50
No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me
– bebe
Nov 22 at 19:06
@TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question
– bebe
Nov 22 at 19:09
That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran
– bebe
Nov 22 at 19:12
This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...)
– TonyK
Nov 22 at 19:15
add a comment |
It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.)
– TonyK
Nov 22 at 18:50
No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me
– bebe
Nov 22 at 19:06
@TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question
– bebe
Nov 22 at 19:09
That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran
– bebe
Nov 22 at 19:12
This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...)
– TonyK
Nov 22 at 19:15
It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.)
– TonyK
Nov 22 at 18:50
It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.)
– TonyK
Nov 22 at 18:50
No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me
– bebe
Nov 22 at 19:06
No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me
– bebe
Nov 22 at 19:06
@TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question
– bebe
Nov 22 at 19:09
@TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question
– bebe
Nov 22 at 19:09
That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran
– bebe
Nov 22 at 19:12
That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran
– bebe
Nov 22 at 19:12
This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...)
– TonyK
Nov 22 at 19:15
This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...)
– TonyK
Nov 22 at 19:15
add a comment |
1 Answer
1
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0
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HINT
To check differentiability let apply the definition and check the existence of
$$lim_{xto 0}frac{f(x)-f(0)}{x-0}$$
For the second point, calculate $f'(x)$ and refer to
The definition of continuously differentiable functions
answered Nov 22 at 18:46
gimusi
92.7k94495
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
HINT
To check differentiability let apply the definition and check the existence of
$$lim_{xto 0}frac{f(x)-f(0)}{x-0}$$
For the second point, calculate $f'(x)$ and refer to
The definition of continuously differentiable functions
answered Nov 22 at 18:46
gimusi
92.7k94495
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
add a comment |
up vote
0
down vote
HINT
To check differentiability let apply the definition and check the existence of
$$lim_{xto 0}frac{f(x)-f(0)}{x-0}$$
For the second point, calculate $f'(x)$ and refer to
The definition of continuously differentiable functions
answered Nov 22 at 18:46
gimusi
92.7k94495
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
add a comment |
up vote
0
down vote
up vote
0
down vote
HINT
To check differentiability let apply the definition and check the existence of
$$lim_{xto 0}frac{f(x)-f(0)}{x-0}$$
For the second point, calculate $f'(x)$ and refer to
The definition of continuously differentiable functions
answered Nov 22 at 18:46
gimusi
92.7k94495
HINT
To check differentiability let apply the definition and check the existence of
$$lim_{xto 0}frac{f(x)-f(0)}{x-0}$$
For the second point, calculate $f'(x)$ and refer to
The definition of continuously differentiable functions
answered Nov 22 at 18:46
gimusi
92.7k94495
edited Nov 22 at 18:51
answered Nov 22 at 18:46
gimusi
92.7k94495
answered Nov 22 at 18:46
gimusi
92.7k94495
answered Nov 22 at 18:46
gimusi
92.7k94495
92.7k94495
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
add a comment |
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
For the second point, they are asking for the integral, not its derivative. What do you think I should do with that?
– bebe
Nov 22 at 19:14
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
Recall that $F’ =f $ and f is continuous.
– gimusi
Nov 22 at 19:26
add a comment |
It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.)
– TonyK
Nov 22 at 18:50
No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me
– bebe
Nov 22 at 19:06
@TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question
– bebe
Nov 22 at 19:09
That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran
– bebe
Nov 22 at 19:12
This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...)
– TonyK
Nov 22 at 19:15