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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