Sequential Criterion for Continuity: A Formal Definition The sequential criterion for continuity states that a function $f$ is continuous at a point $a$...
The sequential criterion for continuity states that a function is continuous at a point if the following two conditions are satisfied:
Limit Property: As the value of approaches from the left and right sides, the limit of as approaches is equal to .
Right-Left Hand Side Limits Matching: The limit of as approaches from the right side and the limit of as approaches from the left side must be equal to the same value.
In other words, is continuous at if both the left and right limit of at exist and are equal to .
Examples:
Continuous function: is continuous at because both the left and right limits exist and are equal to 1.
Discontinuous function: is discontinuous at because the left and right limits are not equal.
Continuous function: is continuous for all real values of because both the left and right limits exist and are equal to .
Additional Notes:
The sequential criterion is a stronger condition than the open/closed definition of continuity.
A function can be continuous at a point without being continuous at every point in its domain.
The sequential criterion can be applied to both single-variable and multi-variable functions