Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Continuity definition Continuous at a point the interior point A function Y equal F of X is continuous at an interior Point C of its domain. If the limit is X approaches C of F of X is the same as the value F at that location, CF of C at an end point. A function Y equal F of X is continuous at a left end point A or at a right end point B of its domain. If the limit as X approaches a from the right of F of X = F of A, this is called right continuous or the limit as X approaches B from the left of F of X equal F of B. This is called left continuous. If a function is not continuous at a Point C, we say that F is discontinuous. At C&C is a point of discontinuity continuity test. A function F of X is continuous X = C if and only if it meets the following conditions. 1F of C must exist to the limit of X approaching C of F of X has to exist, and three the limit as X approaches C of F of X has to equal F of C There are many different discontinuity types. One is removable discontinuity where F of C doesn't exist, but the limit as X approaches C of F of X would exist. A different type is jump discontinuity. This usually occurs in piecewise functions where the left side and the right side are not the same for the limit. Another type is oscillating type where as we get closer and closer and closer it does extreme values. An example of Y equals sine of 3 / X. Still another one is an infinite type of discontinuity where the left side and the right side go to different values on an asymptote on a vertical asymptote. Now we need we do need to clarify that functions can be continuous on an interval versus continuous on a domain. So when we look at this infinite, the domain is negative Infinity to two union, two to Infinity. So it's not continuous on its on the function of real numbers, but it actually is continuous on the domain of the function. If I look at just negative Infinity to two, that is continuous. If I look at 2:00 to Infinity, that's also continuous. But over the real numbers it's not continuous. Over the domain of the function it is continuous. That's an important distinction. Let's look at some examples. Y equal X + 3 / X ^2 -, 3 X -10. We want to know where it's continuous, so we're going to factor that denominator and realize that vertical asymptotes occur at X = 5 and X equal -2. Hence, at X = 5 and X = -2, there's an infinite discontinuity. So the domain would be negative Infinity to -2 union -2 to 5 union five to Infinity. If we're looking at the discontinuities that would occur over the real numbers that would occur at -2 and five, and the type of discontinuity would be an infinite discontinuity. If we look at another example where we're using roots, if we had Y equal the 4th root of 3 X -1 because the index is even, we know the inside would have to be greater than or equal to 0. If I add one divide by three, I get X greater than or equal to 1/3. So this is continuous from 1/3 to Infinity. It has A1 sided continuity at one third because it wasn't defined at the other side of 1/3, only on the positive direction. If we look at Y equal 2 -, X to the one fifth, that's the same thing as the 5th root of 2 -, X because here the index is odd. We know that we can have positive negatives or zeros underneath an odd index. So it's just continuous everywhere, negative Infinity to Infinity. We want to talk about continuous extensions next. So we're given a little H of T being this function, and we want to figure out how to define a capital H of T so that this is always continuous. And what's going to happen is if we factor this, we get t + 5 T -2 / t -, 2. The T minus twos cancel. So this H of T is really t + 5 when T doesn't equal 2. So the capital H of T is going to be little H of T whenever T is not equal to two. But if T was two, it would be undefined. So we're going to look at what it's simplified to and put two in for that. So 2 + 5 would give us a value of seven. So when T equal to, we'd want seven to make this a continuous extension. So it would be defined everywhere. This is really if we thought about looking at its graph. This would be a graph of the line t + 5, but at the value of two, there'd be a hole. So to make this a continuous extension, we need to fill the hole and to fill the hole. When T is 2, our Y value would be 7. If Y was seven, that would fill that hole. That occurred when T equal to. Now it is continuous. This is a very important and powerful tool to be able to take something that's not continuous, add a piece to it, making it a piecewise function and making it continuous. When we look at this next one, we get X for X less than -2. So the Y equal X is really just a line at -2 be an open dot and the slope of one. So it would be going down like this, B times X ^2 where X is greater than or equal to -2. Well, this is a parabola. And what we want to do is we want to figure out how to fill this hole so that then it's going to be continuous. So I need this piece here to be filling that hole with my parabola. So what I want is I want BX squared to actually equal X and we want to solve for B when X equal -2 because I want the left side and the right side to be the same, so it's continuous. So b * -2 ^2 would equal -2 four, B equal -2, B equal, negative 1/2. So if B was negative 1/2, we would actually get a parabola that would go down. And if we thought about -1 half X ^2, we know that the vertex would be 00. But when X is -2 negative 2 ^2 would be 4/4 times negative 1/2 would give us -2. It would share this point here. So this would be a new graph. It would be a piecewise graph that would be a continuous extension because it's now going to be, it's going to have continuity at every point. So we wanted to define B so that this was a continuous graph. B would have to be negative 1/2 for it to be a continuous graph if we wanted to look at some involving trig functions. If we look at limit as X approaches π of sine of X minus sine X and we want to know we want to find the limit and then we want to know is it continuous as the point gets approached? Well, if we put in sine of π here, sine of π is really just zero. So we're going to have π -, 0 and a sine of π is 0 again. So yes, this is going to be continuous and the limit is defined. Thank you and have a wonderful day.