# How do you find the points of continuity of a function?

For functions we deal with in lower level Calculus classes, it is easier to find the points of discontinuity. Then the points of continuity are the points left in the domain after removing points of discontinuity

A function cannot be continuous at a point outside its domain, so, for example:

It is worth learning that rational functions are continuous on their domains.

For functions defined piecewise, we must check the partition number, the points where the rules change. The function may or may not be continuous at those points.

and also these two numbers are equal (a strange phrase, but it is common enough -- I mean these two descriptions pick out the same number.)

Example:

In interval notation:

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To find the points of continuity of a function, you need to check three conditions:

- The function must be defined at that point.
- The limit of the function as x approaches that point must exist.
- The limit of the function as x approaches that point must be equal to the value of the function at that point.

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