for all real numbers at for all real numbers, except impossible to say
Here we see a consequence of a function being continuous.
Continuity of Polynomial Functions All polynomial functions, meaning functions of the form where is a non-negative integer
and each coefficient , , is a real number, are continuous for all real numbers.
Continuity of Rational Functions A rational function h, meaning a function of the form where and are polynomials, is
continuous for all real numbers except where . That is, rational functions are continuous wherever they are defined.
Let be a real
number. Notice that, if , then clearly is not defined (since you are trying to divide by zero). So we will consider only such that .
Then, since is continuous at (since is a polynomial, which we just showed are continuous everywhere), . Therefore, write with me,
and now by the Quotient Law, and by the continuity of polynomials we may now set Since we have shown that , we have shown
that is continuous at .
Where is continuous?
True or false: If and are continuous functions on an interval , then is continuous on .
True False
This means that and .
Now, define a new function, , where , for all in . We have to show that is continuous at , or that
Lat’s start with
and, therefore,
We have proved that is continuous at any number in . Therefore, is continuous on . Similarly, we can prove that is continuous on any interval , by showing it is left-or right-continuous at the endpoints. We can adjust the proof for the function .
True or false: If and are continuous functions on an interval , then is continuous on .
True False
We still don’t know how to compute a limit of a composition of two functions. Our next theorem provides basic rules for how limits interact with composition of functions.
Because the limit of a continuous function is the same as the function value, we can now pass limits to the inside of continuous functions.
Continuity of Composite Functions If is continuous at , and if is continuous at , then is continuous at .
Using the Composition Limit Law, we can compute the last example from the beginning of this section.
Many of the Limit Laws and theorems about continuity in this section might seem like they should be obvious. You may be wondering why we spent an entire section on these theorems. The answer is that these theorems will tell you exactly when it is easy to find the value of a limit, and exactly what to do in those cases.
The most important thing to learn from this section is whether the limit laws can be applied for a certain problem, and when we need to do something more interesting. We will begin discussing those more interesting cases in the next section.
Let’s see some examples!
Can this limit be directly computed by limit laws?
yes no
Compute:
Can this limit be directly computed by limit laws?
yes no