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Discontinuity calculus examples11/1/2023 Identify where the function has a removable discontinuity and determine the value of the function that would make it continuous at that point. Since the value of f is not the same as the limiting value here, we can say that f has a removable discontinuity at x = 2. Because the curve approaches y = 3 from the left and the right, the limit is equal to 3.īut f(2) = 1 (based on the location of the dot). This situation happens in the graph shown below. That is, the limit exists, the function value exists, but they are different values. Suppose both conditions 1 and 2 hold for a function at a given point, but condition 3 fails. That’s the clue that we’re dealing with a removable discontinuity at x = -2. In fact, the graph would be continuous at that point if the hole at (-2, 2) were filled in. If it really is a removable discontinuity, then filling in the hole results in a continuous graph! You might imagine what happens if you filled in that hole. If the limit exists, but f( a) does not, then we might visualize the graph of f as having a “hole” at x = a. Next we’ll discuss what happens if condition 1 holds (the limit exists), but either condition 2 or 3 fail.Ī function f has a removable discontinuity at x = a if the limit of f( x) as x → a exists, but either f( a) does not exist, or the value of f( a) is not equal to the limiting value. ![]() In other words, condition 1 of the definition of continuity failed. In the previous cases, the limit did not exist. The last category of discontinuity is different from the rest. Therefore, f( x) = sin(1/ x) has a discontinuity at x = 0, of the infinite oscillation variety. Then replacing x by 1/ x in the argument has the effect of taking all those infinitely many periodic waves of the sine function as x → ∞ and squeezing them next to the origin instead.Īt any rate, since there is no single value of y to which the curve seems to be heading as x → 0, the limit does not exist at x = 0. The reason for this strange behavior has to do with the fact that sin x itself is periodic. On the other hand, as x approaches 1 from the right, the values of y seem to get closer and closer to y = 3. In the graph shown below, there seems to be a “mismatch.” As x approaches 1 from the left, that part of the graph seems to land on y = -1. One way in which a limit may fail to exist at a point x = a is if the left hand limit does not match the right hand limit. Then, depending on how the limit failed to exist, we classify the point further as a jump, infinite, or infinite oscillation discontinuity. If the limit as x → a does not exist, then we can say that the function has a non-removable discontinuity at x = a. A limit may fail to exist for a variety of reasons. The first condition, that the limit must exist, is especially interesting. When one or more of these conditions fails, then the function has a discontinuity at x = a, by definition. So, the number L that you get by taking the limit should be the same value as f( a). The limit must agree with the function value.Think of this equation as a set of three conditions. So let’s begin by reviewing the definition of continuous.Ī function f is continuous at a point x = a if the following limit equation is true. A function is discontinuous at a point x = a if the function is not continuous at a. The definition of discontinuity is very simple. ![]() In fact, there are various types of discontinuities, which we hope to explain in this review article. ![]() ![]() What’s a discontinuity? Any point at which a function fails to be continuous is called a discontinuity.
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