In this lesson, our instructor Vincent Selhorst-Jones goes over Continuity and One-sided Limits. This lesson begins with a motivating example to get your started and then explores the definition of continuity. Youll also learn about one-sides limits and the limits of piecewise functions. The lesson ends with four worked-out examples.
Long ago in this course, we learned about continuous functions. At the time, we lacked the formal ideas to precisely define continuity, so we intuitively defined it as being any of these three equivalent things:
All the parts of the function are connected;
The function's graph can be drawn without ever having to lift your pencil from the paper.
There are no "breaks"/"holes" in the graph.
Now we have the the formal background to expand on this intuitive definition and define continuity precisely. Formally, a function f(x) is continuous at c if f(c) exists and
f(x) = f(c).
This is the same as the above intuitive ideas, because when a limit matches up with its function, it means that the function goes where we "expect"-there are no jumps or weird stuff.
If f(x) is not continuous at c, we say it is discontinuous at c. We call such a location a discontinuity. If f(x) is continuous at every point in an interval (a,b), we say f is continuous on (a,b). If f(x) is continuous for
every real number, we simply say that f is a continuous function.
Notice that the vast majority of the functions we're used to working with are continuous. This is why "normal" functions allow us to simply plug in the value we're approaching. Because we generally have
"Normal" ⇒ Continuous ⇒
f(x) = f(c)
Furthermore, this also explains why a function that is "weird" in one place still allows us to plug in the value we're approaching if it's not the "weird" value. Because, other than the "weird" value, the rest of the function is continuous, so we can
apply the same logic.
If a limit does not exist because the two sides do not agree on a single value, we can consider a one-sided limit: the limit of a function as it approaches from only one side.
denotes the limit as x approaches c from the left or negative side. It is the limit as x→ c while x < c.
denotes the limit as x approaches c from the right or positive side. It is the limit as x→ c while x > c.
For one-sided limits, be careful to keep track of which symbol goes with which side; they can be easy to get confused at first. Remember, `−' means look at the `negative' side, while `+' means look at the `positive' side.
If both sides of a function go to the same value as a location is approached, then the normal limit exists there. Furthermore, if a normal limit exists, both sides must go to the same value as they approach it.
f(x) = L =
f(x) = L
Similarly, if both one-sided limits do not go to the same value (or one side doesn't exist), then a normal limit does not exist there.
The above idea allows us to find the limits of piecewise functions at "breakover" points. (If it's not a breakover, we can usually just plug the location in, as discussed in the previous lesson.) We find the limit of a piecewise function by checking
if the left- and right-side limits agree with each other.
If they agree, then the limit exists and equals them.
If they do not agree, then the limit does not exist.
Continuity & One-Sided Limits
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.