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.
This brings up a general principle: a function that has a denominator is not defined (and hence, not continuous) at points where the denominator is
This include "hidden" denominators as we have in
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.
Recall that in order for
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.)
It is important and relevant for piecewise function, to remember that in order for
The domain of
Each of the 4 functions is continuous on the interval on which it is used:
We need to check for continuity at the numbers 2, 7, and 9. Note first (if you have not already done so) that each is in the domain of
Because these are equal, we have
In interval notation: