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.