Single Point Continuity - Spivak Ch.6 Q5

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Miguel said:
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\,?##
 
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##.
 
@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 ;). .
 
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