Question

What are continuous functions?

Answer 1

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