## Continuous Function

A continuous function is a Function where the pre-image of every Open Set in is Open in . A function in a single variable is said to be continuous at point if

1. is defined, so that is in the Domain of .

2. exists for in the Domain of .

3. ,

where lim denotes a Limit. If is Differentiable at point , then it is also continuous at . If and are continuous at , then

1. is continuous at .

2. is continuous at .

3. is continuous at .

4. is continuous at if and is discontinuous at if .

5. is continuous at , where denotes , the Composition of the functions and .