What are continuous functions?

1 Answer
Sep 17, 2015

Answer:

We may also state two alternative definitions of continuous functions, using either the sequential criterion or else using topology and open sets.

Explanation:

Alternative definition number 1
Let #f: X ->Y# be a function and let #(x_n)# be a sequence in X converging to an element x in X, ie #lim(x_n)=x in X#
Then f is continuous at x iff and only if the sequence of function values converge to the image of x undr f, ie #iff lim (f(x_n))=f(x) in Y#

Alternative definition number 2
Let #f: X ->Y# be a function. Then f is continuous if the inverse image maps open subsets of Y into open subsets in X.
ie, #AAA_(open)subeY =>f^(-1)(A) # is open in X