Single Point Continuity - Spivak Ch.6 Q5

[Miguel]

Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :)

https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5

 

[fresh_42]

Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :)

https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5

The idea is: No matter how close ($\delta$) you get to $x=a$, there is always a positive distance ($\varepsilon$) such that there are image points ($f(a),f(x)$) at least so far from another.

Can you formally negate the definition of continuity, i.e. formally define what it means to be discontinuous at $a\,?$

 

[Mark44]

@Miguel, in future posts, please do not delete the homework template. Its use is required here for homework questions.

 

[hilbert2]

There are many variations of this problem, another one is

$f(x) = \left\{\begin{array}{c}x,\hspace{20pt}x\in\mathbb{Q}\\-x,\hspace{20pt}x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right.$

which is continuous only at $x=0$.

 

[WWGD]

@Miguel: There is also a notion of continuity through sequences and their convergence*: f is continuous at x iff $( x_n \rightarrow x) \rightarrow (f(x_n) \rightarrow f(x))$ Can you see what happens as you approach a Rational or Irrational through a sequence? What happens when x=a? You may also want to consider the open set definition: Consider an open set in the target space (ban open interval). What is its inverse image under this map. Is ihe inverse image open **?*This is not valid for all spaces, but it is for this one
**This is really not the pointwise definition, but I think it is a nice exercise.

EDIT: Miguel: We don't allow division by 0 in the Math section. You must get an upvote quickly to avoid the 1/0 in your Avatar ;). .