Continuous function
This page lists a core term of calculus. The term is used widely, and a thorough understanding of its definition is critical.
See a complete list of core terminology
Contents
Definition for functions of one variable
At a point
Consider a function and a real number such that is defined in an open interval containing , i.e., is defined at and on the immediate left and right of . We say that is continuous at if it satisfies the following equivalent definitions:
No. | Shorthand | What the definition says |
---|---|---|
1 | in terms of limits | . In words, the limit of as exists and equals the value of the function at . |
2 | in terms of one-sided limits | . In words, the left hand limit of at , the right hand limit of at , and the value of at are all equal. |
3 | in terms of left and right continuity | is both left and right continuous at . |
4 | For every , there exists such that for all satisfying (i.e., ), we have (i.e., ). | |
4' | (variant) | For every , there exists such that for all satisfying (i.e., ), we have (i.e., ). |
5 | in terms of centered open balls (same as without the symbols) | For every open ball (i.e., open interval) centered at , there is an open ball (i.e., open interval) centered at such that the image of the open ball centered at lies inside the open ball centered at . [SHOW MORE] |
6 | in terms of not necessarily centered open balls | For every open ball (i.e., open interval) containing , there is an open ball containing such that the image of the open ball containing lies inside the open ball containing . |
Definition of one-sided continuity
Left continuity: Consider a function and a real number such that is defined at and on the immediate left of . We say that is left continuous at if the left hand limit of at exists and equals , i.e., .
Right continuity: Consider a function and a real number such that is defined at and on the immediate right of . We say that is right continuous at if the right hand limit of at exists and equals , i.e., .
On an interval
Consider an interval, which may be open or closed at either end, and may stretch to on the left or on the right. A function from such an interval to the real numbers is termed continuous if it satisfies the following two conditions:
- It is continuous (in the sense of continuous at a point) at all points in the interior of the interval, i.e., all points such that there is an open ball containing the point lying inside the domain interval.
- It has the appropriate one-sided continuity at endpoints: If the interval has a left endpoint (e.g., the interval is of the form , , or , then it must be right continuous at the left endpoint ( in all three example intervals). If the interval has a right endpoint (e.g., the interval is of the form , , or ), then it must be left continuous at the right endpoint ( in all three example intervals).