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