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How to prove that this function is differentiable?

How to know this function is continuous and differentiable?Prove that $f$ is differentiable at $(x)=0$ using formal definitionproving that a certain function is not differentiable at $(0,0)$Showing that this function is infinitely differentiableIs this function differentiable at the origin?How do I prove this function is differentiable at 0?Is this function differentiable in $(1,-1)$?Prove that a function is not continuously differentiableProve this function is differentiable at a point but the partial derivatives are not continuousProve function is not differentiable even though all directional derivatives exist and it is continuous.

1

$begingroup$

Let $$f(x,y)=begincases(x^2+y^2) sinleft(frac1x^2+y^2right)&textif;x^2+y^2 neq 0\\0&textotherwise endcases$$

Prove that $f$ is differentiable

Could anyone give me a hint please?

calculus multivariable-calculus derivatives

share|cite|improve this question

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

$endgroup$

add a comment | 

1

$begingroup$

Let $$f(x,y)=begincases(x^2+y^2) sinleft(frac1x^2+y^2right)&textif;x^2+y^2 neq 0\\0&textotherwise endcases$$

Prove that $f$ is differentiable

Could anyone give me a hint please?

calculus multivariable-calculus derivatives

share|cite|improve this question

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

$endgroup$

add a comment | 

$begingroup$

Let $$f(x,y)=begincases(x^2+y^2) sinleft(frac1x^2+y^2right)&textif;x^2+y^2 neq 0\\0&textotherwise endcases$$

Prove that $f$ is differentiable

Could anyone give me a hint please?

calculus multivariable-calculus derivatives

share|cite|improve this question

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

$endgroup$

share|cite|improve this question

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

share|cite|improve this question

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

edited Mar 12 at 10:48

Chinnapparaj R

5,7332928

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

asked Mar 12 at 10:43

IntuitionIntuition

1,108826

2 Answers
2

active

oldest

votes

1

$begingroup$

Hint: Prove that$$lim_(x,y)to(0,0)fracf(x,y)-f(0,0)lVert(x,y)rVert=0.$$

share|cite|improve this answer

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what about not (0,0)?
  $endgroup$
  – Intuition
  Mar 12 at 11:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why we do this?
  $endgroup$
  – Intuition
  Mar 12 at 11:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Outside $(0,0)$, your function can be expressed from differentiable functions using arithmetic operations and composition. Therefore, it is differentiable there.
  $endgroup$
  – José Carlos Santos
  Mar 12 at 11:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We do this in order to prove that $f'(0,0)$ is the null function.
  $endgroup$
  – José Carlos Santos
  Mar 12 at 11:05
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why we want to prove this?
  $endgroup$
  – Intuition
  Mar 12 at 11:07
  

  
  
  
  

 | 
show 5 more comments

1

$begingroup$

Hint: Use the polar coordinates
$$x = r costheta \ y = r sin theta$$

and simplify $f(x,y) to f(r, theta)$.

share|cite|improve this answer

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519

$endgroup$

add a comment | 

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1

$begingroup$

Hint: Prove that$$lim_(x,y)to(0,0)fracf(x,y)-f(0,0)lVert(x,y)rVert=0.$$

share|cite|improve this answer

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what about not (0,0)?
  $endgroup$
  – Intuition
  Mar 12 at 11:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why we do this?
  $endgroup$
  – Intuition
  Mar 12 at 11:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Outside $(0,0)$, your function can be expressed from differentiable functions using arithmetic operations and composition. Therefore, it is differentiable there.
  $endgroup$
  – José Carlos Santos
  Mar 12 at 11:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We do this in order to prove that $f'(0,0)$ is the null function.
  $endgroup$
  – José Carlos Santos
  Mar 12 at 11:05
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why we want to prove this?
  $endgroup$
  – Intuition
  Mar 12 at 11:07
  

  
  
  
  

 | 
show 5 more comments

1

$begingroup$

Hint: Prove that$$lim_(x,y)to(0,0)fracf(x,y)-f(0,0)lVert(x,y)rVert=0.$$

share|cite|improve this answer

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what about not (0,0)?
  $endgroup$
  – Intuition
  Mar 12 at 11:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why we do this?
  $endgroup$
  – Intuition
  Mar 12 at 11:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Outside $(0,0)$, your function can be expressed from differentiable functions using arithmetic operations and composition. Therefore, it is differentiable there.
  $endgroup$
  – José Carlos Santos
  Mar 12 at 11:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We do this in order to prove that $f'(0,0)$ is the null function.
  $endgroup$
  – José Carlos Santos
  Mar 12 at 11:05
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why we want to prove this?
  $endgroup$
  – Intuition
  Mar 12 at 11:07
  

  
  
  
  

 | 
show 5 more comments

$begingroup$

Hint: Prove that$$lim_(x,y)to(0,0)fracf(x,y)-f(0,0)lVert(x,y)rVert=0.$$

share|cite|improve this answer

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

share|cite|improve this answer

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

answered Mar 12 at 10:57

José Carlos SantosJosé Carlos Santos

168k22132236

1

$begingroup$

Hint: Use the polar coordinates
$$x = r costheta \ y = r sin theta$$

and simplify $f(x,y) to f(r, theta)$.

share|cite|improve this answer

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519

$endgroup$

add a comment | 

1

$begingroup$

Hint: Use the polar coordinates
$$x = r costheta \ y = r sin theta$$

and simplify $f(x,y) to f(r, theta)$.

share|cite|improve this answer

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519

$endgroup$

add a comment | 

$begingroup$

Hint: Use the polar coordinates
$$x = r costheta \ y = r sin theta$$

and simplify $f(x,y) to f(r, theta)$.

share|cite|improve this answer

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519

$endgroup$

share|cite|improve this answer

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519

answered Mar 12 at 11:03

Piotr BenedysiukPiotr Benedysiuk

1,344519