Calculus I, Chapter 3: The Average Rate of Change and Limits
Previous Section: 3.5 Limits, Graphically
In the previous sections we identified that a function whose one-sided limits are equal tends to look like the lines of the function are going towards each other. In a way, it looks like the function does not have any breaks in it, even if it does. There is a name for when a function’s values and limits are equal: Continuity.
Definition
Continuity at a point describes a function where the following three things occur at some point x=a:
lim┬(x→a) f(x) exists,f(a) exists, lim┬(x→a) f(x)=f(a)
We can extend this to talk about a function as a whole by saying that these three conditions hold true on the entire domain of a function. If that is the case, we call a function continuous. A simple way to identify continuity is to use the graph, if you can draw the function without picking up your pencil then it is probably continuous.
A more formal definition is the following:
[formal def]
We need to remember how we got here for a moment. We are attempting to understand the instantaneous rate of change of a function. This definition is improved by using a limit, so we need to understand what limits are and how to solve them. Why is continuity important for limits? The reason is that evaluating the limit of a continuous function is easy. This is because if a function is continuous at a point, we automatically know that the limit exists and is equal to the function itself. So, to evaluate the limit of a continuous function we need only plug into the function. This applies both on intervals of continuity and points of continuity.
Example 1: Show that f(x)=x^2+2x is continuous at x=2.
Solution:
First, let’s evaluate the limit:
| x | L |
| 1.9 | |
| 1.99 | |
| 1.999 | |
| 2 | |
| 2.001 | |
| 2.01 | |
| 2.1 |
Second, we evaluate the function: f(2)=2^2+2(2)=8.
Finally, Since lim┬(x→2) f(x) exists and is equal to f(2) we know that the function is continuous at x=2.
Again, to evaluate the limit of a continuous function we need only plug into the function. In other words, the only thing that affects the values of a limit are discontinuities.
Types of Discontinuity
A discontinuity is a point at which a function is not continuous. There are three main forms of discontinuity: Holes, asymptotes, and jumps. These are helpful to know and identify because if a function does not contain one of these discontinuities, then it must be continuous!
A hole in a graph, more formally called a removable discontinuity, is a point at which the function has the value 0/0 when we plug in the x-value. However, a hole does not affect where the graph points to, so this does not affect the value of a limit.
An asymptote is more often referred to as an infinite discontinuity in this context. This is because the function usually approaches infinity on one or both sides of a vertical asymptote. These occur when the function has a value that is divided by zero, but is not zero in the numerator, like (1/0). In these discontinuities, it may be the case that the limit exists, if both sides go to the same value. However, it may also be the case that the limit does not exist, if the sides go to different values.
The final type of discontinuity is the jump discontinuity. Sometimes referred to as breaks, these are usually a part of a piecewise function. This is because the sides of the function are disjointed and don’t connect. It appears as if the function ‘jumped’ to another spot.
Example 2: Identify the discontinuity of f(x)= (x-2)/(x+3)
Solution:
Since (x-2)/(x+3) as a vertical asymptote at x+3=0 we know that there is an infinite discontinuity at x=-3.
Example 3: Identify the discontinuities of g(x)=cos((x-4)/(x^2-12x+32))
Solution:
For cos((x-4)/(x^2-12x+32)), the cos() can be ignored because it doesn’t have any domain restrictions inherent to it, so the function can be rewritten as (x-4)/(x^2-12x+32) in order to find discontinuities.
(x-4)/(x^2-12x+32)=(x-4)/((x-4)(x-8))
Thus, there are two discontinuities. x=4 is a removable discontinuity and x=8 is an infinite discontinuity.
Review
A function, f(x), is continuous at some point x=a if lim┬(x→a) f(x) exists,f(a) exists,and lim┬(x→a) f(x)=f(a)
There are three types of discontinuity: Removable (holes), Infinite (asymptotes), Jump (piecewise functions)