Practical lesson. The study of function continuity and classification of discontinuity points
Our practical class is devoted to the study of function continuity and classification of discontinuity points.
Let's take some examples. The function f is set by different formulas at different intervals. What do we pay attention to? Generally speaking, if written separately, all three formulas: minus x minus 3, one divided by x + 2, x minus 1 squared are elementary functions that are continuous in their domains.
How can we link the function f to these functions? How can we use what we know about the continuity of these functions?
We are going to use the following theorem: if two functions coincide on a certain interval, and one of the functions on this interval is continuous, then the other function is continuous on this interval, too.
By the way, you can try to prove this fact.
If there is no interval, but a segment, that is, we include at least one of the points into this interval, then the theorem ceases to be true.
So please note: we are talking about the coincidence at intervals.
How are we going to use this fact?
Let’s return to the function f.
The elementary functions (they are listed here) are continuous within their domains, that is, at any point in the domain, and therefore on intervals. What intervals does the function f coincide with these functions on?
It coincides not only on intervals, but also at the point -2, but we can apply the theorem only on intervals, so we specify the largest intervals on which the function f coincides with the specified functions.
It coincides with the first function on the interval from minus infinity to 2, with the second function - on the interval from minus 2 to 0, and with the third function – on the interval from 0 to plus infinity. Thus, we already know a huge class of points where the function f is continuous.
All we have to do is to find out: is the function f continuous or not at points minus 2 and 0? Let's consider these points.
Let's write down the function f so that we can remember what it looks like.
Let's also write down the first point -2. This is the point at which the function f passes from one formula to another. So, -2. What are we going to use? We are going to use the definition of continuity.
The function is continuous if the limit of the function at a point is equal to the value of the function at that point.
We are to calculate one-sided limits. On the left, the function f is set to the left of the point -2 (we consider the first formula minus x minus 3 which is to the left of the point -2). The limit of this function at the point -2 is minus 1. Thus, the limit to the left of this function is minus 1.
Now, we calculate what is on the right. To the right of the point -2. The function f is set by a completely different formula. The value is calculated using the formula: 1 divided by x plus 2. We plug in -2, and get 1 divided by 0. In the theory of limits, this is infinity. You can also take into account the infinity sign. x is greater than -2, so the denominator has a + sign, greater than 0, the fraction is always positive. Although the sign is not important here.
What is the most important here? At least one of the one-sided limits at point -2 is equal to infinity.
If at least one of the one sided limits at the point x0 is equal to infinity or does not exist, then x0 is a break point of the second kind.
Thus, we have the first conclusion. -2 is a break point of the second kind for the given function f.
Let us consider the next point x2 that is equal to 0. We calculate one-sided limits. x approaches 0 from the left, that is, it is still less than 0, it is to the left of the point 0. The function is set by the formula 1 divided by x + 2. We plug in 0 and obtain ½.
On the right (to the right of the point 0) the function is set as x minus 1 squared. We calculate the limit by plugging 0 instead of x, and we get 1. We see that both one-sided limits are finite, but not equal.
This means that there is no limit at the point 0.
If one-sided limits are finite, then this is a first-kind break. However, if these limits are not equal, then this break is irremediable. We speak of a function jump in this case, a jump from 1/2 to 1.
Let us look at the graph of this function. At the point -2, the limit on the right is plus infinity. It is a jump; or rather, a break of the second kind, but at the point 0 there is a jump.
To draw a picture, we need the hand to jump up from 1/2 to 1 (it is a jump), this is a break of the first kind, the jump does not eliminate the break.
Let's take the second example. The function is set by two formulas at the intervals from minus infinity to 2, from 2 to plus infinity, and at the point 2 the value is said to be 5.
Let's try to figure out where the function is continuous, and whether it has any break points.
The function f coincides with a quadratic function and a linear function on two intervals, so the function is continuous on these intervals. We only need to deal with the point 2.
x0 is the only point that requires some clarification.
We calculate one-sided limits. To the left of the point 2 it is a quadratic function. We substitute, calculate, and get 5. On the right, we plug 2 in the expression for a linear function, and again we have 5. Thus, the one-sided limits are finite.
This means that the limit of the function at the point 2 is equal to this general value. Thus, the limit of the function at the point 2 exists, it is finite, and is equal to 5.
So when is the function continuous?
If the function value is equal to the limit, the limit and value must be equal.
According to our calculation the limit of the function f is equal to 5 and the value at the point 2 is also 5.
Thus, the limit and the function value are equal, then by the definition of continuity the function f at the point 2 is also continuous. The conclusion: the function f is continuous at the point 2, which means it is continuous on R (we could just say continuous, since R is its domain).
If we look at the graph of this function, we see that we can build the entire line without taking off our hands – the graph is a continuous curve