[FOM] A question concerning continuous functions

[Ayan Mahalanobis]

This is some comments on the recent posting on Arnon Avron.

I also had similar problems with the epsilon-delta definition of 
continuity and later bought it as a religion as the whole limit process 
is, unjustifiable but working. The best defense about the definition of 
continuity is that, a continuous function cannot have any jump 
discontinuity", so do I define jump discontinuity first or continuity. 
Also  it is probably not a very good idea to appeal to intuition in a negative way. 

The root of the problem behind epsilon-delta definition is the lack of 
continuity in the definition. It seems to be a reasonable worry to me as 
Brouwer, Bishop and Church et al either proved in their system that all 
functions are continuous or worked only with continuous functions.

I think the definition is so successful because classically we 
conceive real line as a set of points with the law of trichotomy, which 
makes it ** discrete ** but uncountable. Probably we need to look at our 
classical understanding of continuum more closely. 

Some thoughts, any comments welcome.

Cheers
Ayan