![]() Points over the interval, that the limit as x approachesĪny one of these points of f of x is equal to If you wanted to do more rigorously and you actually had theĭefinition of the function, you might be able to do a proof, that for any of these Look, if I start here, I can get all the way to negative five without having to pick up my pencil. Not-so-mathematically-rigorous way, where you could say, hey, Is f continuous over that interval? Let's see, we're going from negative seven to negative five, and there's a couple of So let's say we're talkingĪbout the open interval from negative seven to negative five. So let's do a couple of examples of that. This open interval, if and only if, if and only if, f is continuous, f is continuous over every point in, over every point in the interval. Points between x equals a and x equals b, but not equaling xĮquals a and x equals b. So the parentheses instead of brackets, this shows that we're not So we say f is continuous over an open interval from a to b. Talk about an open interval, and then we're gonna talkĪbout a closed interval because a closed interval getsĪ little bit more involved. Let me delete this really fast, so I have space to work with. So with that out the way, let's discuss continuity over intervals. To pick up your pencil, this notion of connectedness, that you don't have any jumps or any discontinuities of any kind. Rigorous way of describing this notion of not having Without picking up my pen, well, the value of theįunction at that point should be the same as the limit. Well, in order for theįunction to be continuous, if I had to draw this function So if we approach, if we approach from the left, we're getting to this value. The limit as x approaches c of f of x, so let's say that f of x as x approaches c is approaching some value. And when we first introduced this, we said, hey, this looksĪ little bit technical, but it's actually pretty intuitive. Two-way arrows right over here, the limit of f of x as x approaches c is equal to f of c. ![]() So we say that f is continuous when x is equal to c, if and only if, so I'm gonna make these But to do that, let's refresh our memory about continuity at a point. That means for a continuous function, we can find the limit by direct substitution (evaluating the function) if the function is continuous at a.Going to do in this video is explore continuity over an interval. ![]() In mathematical notation we would write this as: This value of 5 is then called the limit (L) of the function. To calculate the limit as x approaches 3, we ask the question:Īs the x-value of the function gets closer and closer to 3 (but not equal to 3), what value does the y-value of the function get closer and closer to ? From the graph we can determine that the y-value gets closer and closer to the value of 5. To calculate the limit of this function as x approaches c, we ask the question:Īs the x-value of the function gets closer and closer to c (but not equal to c), what value does the y-value of the function get closer and closer to ? This result is called the limit (L) of the function.įrom the graph we know that the point (3, 5) is not defined for this function. Notice in Figure 2.52, the open circle at the point (c, L) indicates the function is not defined at this point. To find the limit of a function f(x) (if it exists), we consider the behavior of the function as x approaches a specified value. The limit of a function describes the behavior of the function when the variable is near, but does not equal, a specified number ( Figure 2.42).
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