# using either the sequential criterion or else using topology and open sets.

using either the sequential criterion or else using topology and open sets.

What are continuous functions?

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

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