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Replace the epsilon and delta in the definition of continuity, what kind of continuity do we get?

The epsilon-delta definition of continuityThe epsilon-delta definition of continuity$delta$ and $epsilon$ in the continuity definitionEpsilon-delta definition of continuityWhy epsilon and delta can be changed in the definition of continuity when the function is limited in an intervalCan limits be defined in a more algebraic way, instead of using the completely analytic $delta$-$epsilon$ definition?Test the Function for continuity using the Epsilon-Delta definition.Show continuity of a function using epsilon delta definitionHow to find $delta$ in the $varepsilon-delta$ definition of continuityDelta Epsilon Continuity proof exampleProof of equivalence between open-set and $varepsilon$-$delta$ definition of continuity.

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1

$begingroup$

1) $f : X to Y$ is a map. For all $x_0in X$, if for every $delta > 0$, there is $varepsilon > 0$ such that $d_X(x_0, x) < delta$ whenever $d_Y(f(x_0), f(x)) < varepsilon$.

2)$f : X to Y$ is a map. For all $x_0in X$, if for every $ varepsilon> 0$, there is $delta > 0$ such that $d_Y(f(x_0), f(x)) < delta$ whenever $d_X(x_0, x) < varepsilon$.

Is map $f$ continuous in the two cases? If not, what kind of continuity is this?

For the second the case, I guess this definition is much stronger than continuity?

real-analysis calculus analysis functions continuity

share|cite|improve this question

edited Mar 21 at 7:26

High GPA

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You have shown that $f^-1$ is continuous at $f(x_0)$ (if it exists at all) instead of showing that $f$ is continuous at $x_0.$
  $endgroup$
  – Dbchatto67
  Mar 21 at 6:57
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The function is bounded in the second case
  $endgroup$
  – vidyarthi
  Mar 21 at 7:24
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Dbchatto67 Due to my humble love of math
  $endgroup$
  – High GPA
  Mar 21 at 7:27
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  why the downvote?
  $endgroup$
  – vidyarthi
  Mar 21 at 7:28
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  see here
  $endgroup$
  – vidyarthi
  Mar 21 at 7:30
  

  
  
  
  

 | 
show 2 more comments

0

1

$begingroup$

1) $f : X to Y$ is a map. For all $x_0in X$, if for every $delta > 0$, there is $varepsilon > 0$ such that $d_X(x_0, x) < delta$ whenever $d_Y(f(x_0), f(x)) < varepsilon$.

2)$f : X to Y$ is a map. For all $x_0in X$, if for every $ varepsilon> 0$, there is $delta > 0$ such that $d_Y(f(x_0), f(x)) < delta$ whenever $d_X(x_0, x) < varepsilon$.

Is map $f$ continuous in the two cases? If not, what kind of continuity is this?

For the second the case, I guess this definition is much stronger than continuity?

real-analysis calculus analysis functions continuity

share|cite|improve this question

edited Mar 21 at 7:26

High GPA

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You have shown that $f^-1$ is continuous at $f(x_0)$ (if it exists at all) instead of showing that $f$ is continuous at $x_0.$
  $endgroup$
  – Dbchatto67
  Mar 21 at 6:57
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The function is bounded in the second case
  $endgroup$
  – vidyarthi
  Mar 21 at 7:24
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Dbchatto67 Due to my humble love of math
  $endgroup$
  – High GPA
  Mar 21 at 7:27
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  why the downvote?
  $endgroup$
  – vidyarthi
  Mar 21 at 7:28
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  see here
  $endgroup$
  – vidyarthi
  Mar 21 at 7:30
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

1) $f : X to Y$ is a map. For all $x_0in X$, if for every $delta > 0$, there is $varepsilon > 0$ such that $d_X(x_0, x) < delta$ whenever $d_Y(f(x_0), f(x)) < varepsilon$.

2)$f : X to Y$ is a map. For all $x_0in X$, if for every $ varepsilon> 0$, there is $delta > 0$ such that $d_Y(f(x_0), f(x)) < delta$ whenever $d_X(x_0, x) < varepsilon$.

Is map $f$ continuous in the two cases? If not, what kind of continuity is this?

For the second the case, I guess this definition is much stronger than continuity?

real-analysis calculus analysis functions continuity

share|cite|improve this question

edited Mar 21 at 7:26

High GPA

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421

$endgroup$

share|cite|improve this question

edited Mar 21 at 7:26

High GPA

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421

share|cite|improve this question

edited Mar 21 at 7:26

High GPA

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421

asked Mar 21 at 6:53

High GPAHigh GPA

1,008421