Some basic facts about continuity

Reference: (1) Topology by Hocking and Young especially chapter 1 (2) Analysis — Walter Rudin.

Definition 1: A transformation f: S \rightarrow T is continuous provided that if p is a limit point of a subset X of S, then f(p) is a limit point or a point of f(X).

Definition 2: We may also state the continuity requirement on f as follows: if p is a limit point of \overline{X}, then f(p) is a point of \overline{f(X)}.

Theorem 1: If S is a set with the discrete topology and f: S \rightarrow T any transformation of S into a topologized set T, then f is continuous.

Theorem 2: A real-valued function y=f(x) defined on an interval [a,b] is continuous provided that if a \leq x_{0} \leq b and \epsilon >0, then there is a number \delta >0 such that if |x-x_{0}|<\delta, x in [a,b], then |f(x) - f(x_{0})|< \epsilon. (NB: this is same as definition 1 above).

Theorem 3: Let f: S \rightarrow T 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 f^{-1}(O) is open in S.

Theorem 4:

A necessary and sufficient condition that the transformation f: S \rightarrow T 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 f(x), then there is an open set, U in S containing x and such that f(U) lies in V.

Theorem 5:

A one-to-one transformation f: S \rightarrow T of a space S onto a space T is a homeomorphism, if and only if both f and f^{-1} are continuous.

Theorem 6:

Let f: M \rightarrow N be a transformation of the metric space M with metric d into the metric space N with metric \rho. A necessary and sufficient condition that f be continuous is that if \epsilon is any positive number and x is a point of M, then there is a number \delta >0 such that if d(x,y)< \delta, then \rho(f(x), f(y)) < \epsilon.

Theorem 7:

A necessary and sufficient condition that the one-to-one mapping (that is, a continuous transformation) f: S \rightarrow T of the space S onto the space T be a homeomorphism is that f is interior. (NB: A transformation f: S \rightarrow T 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).

Regards,

Nalin Pithwa.

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