Reference: (1) Topology by Hocking and Young especially chapter 1 (2) Analysis — Walter Rudin.
Definition 1: A transformation is continuous provided that if p is a limit point of a subset X of S, then is a limit point or a point of .
Definition 2: We may also state the continuity requirement on f as follows: if p is a limit point of , then is a point of .
Theorem 1: If S is a set with the discrete topology and any transformation of S into a topologized set T, then f is continuous.
Theorem 2: A real-valued function defined on an interval is continuous provided that if and , then there is a number such that if , x in , then . (NB: this is same as definition 1 above).
Theorem 3: Let be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image is open in S.
A necessary and sufficient condition that the transformation of the space S into the space T be continuous is that if x is a point of S, and V is an open subset of T containing , then there is an open set, U in S containing x and such that lies in V.
A one-to-one transformation of a space S onto a space T is a homeomorphism, if and only if both f and are continuous.
Let be a transformation of the metric space M with metric d into the metric space N with metric . A necessary and sufficient condition that f be continuous is that if is any positive number and x is a point of M, then there is a number such that if , then .
A necessary and sufficient condition that the one-to-one mapping (that is, a continuous transformation) of the space S onto the space T be a homeomorphism is that f is interior. (NB: A transformation of the space S into the space T is said to be interior if f is continuous and if the image of every open set subset of S is open in T).