Let be a function define from topological space to topological space , then is said to be continuous at a point if every neighbourhood of , there exist a neighbourhood of , such that .
In other words, let be two topological spaces. A function is said to be continuous at a point if and only if for every open set , which contains , there exist an open set , such that ; is the inverse image of .
It can also be defined as, let and be topological spaces. A function is said to be a continuous function at a point of if for any neighbourhood of in , there is a neighbourhood of in such that . The function is said to be a continuous function on if it is continuous at each point of .
Note: It may be noted that a function from topological space to topological space is said to be continuous on if is continuous of each points of .
Theorems:


A function from one topological space into another topological space is continuous if and only if for every open set in , is open in .

A function from one topological space into another topological space is continuous if and only if for every closed set in , is closed in .

If and are topological spaces, then a function is continuous on if and only if for any sub set of , .

If and are topological spaces, then a function is continuous on if and only if for any sub set of , .

If is an arbitrary topological space and is an indiscrete topological space, then every function is a continuous function on .

Let and be topological spaces. If and are continuous mappings, then is continuous.
