Video Transcript
Consider the function 𝑓 of 𝑥 is equal to two 𝑥 plus four when 𝑥 is less than or equal to three. And 𝑓 of 𝑥 equals 𝑥 squared minus three when 𝑥 is greater than three. What type of discontinuity occurs at 𝑥 equals three? a) A jump discontinuity. b) An infinite discontinuity. c) A removable discontinuity. Or d) An oscillating discontinuity.
What do we know about continuity of a function at a number. A function 𝑓 is said to be continuous at a number 𝑎 if the following conditions are satisfied. 𝑓 of 𝑎 must exist. The limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 must exist. And finally, the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 must be equal to 𝑓 of 𝑎. If any of these three conditions are not met, then the function is discontinuous. And 𝑎 is called a point of discontinuity.
First, we want to check these three conditions for our function. We want to consider if there is a discontinuity at 𝑥 equals three and what type of discontinuity it would be. So first, we want to check that 𝑓 of 𝑎 exists. For us, does the function exist at 𝑥 equals three. At 𝑥 equals three, we have two 𝑥 plus four. Two times three plus four equals 10. So the first condition is satisfied. Before we check the next two conditions, let’s sketch this function.
We know at 𝑥 equals three, 𝑦 equals 10. And when 𝑥 is less than three, our function is two 𝑥 plus four. This is our graph from 𝑥 equals three to the left. When 𝑥 is greater than three, the function becomes 𝑥 squared minus three. If we plug three into 𝑥 squared minus three, we get three squared minus three. Nine minus three is six. So we’ll draw a point that is not filled in. And then, we’ll sketch the exponential function to the right of 𝑥 equals three.
Back to our conditions for continuity, does the limit as 𝑥 approaches 𝑎 for 𝑓 of 𝑥 exist? As we approach 𝑥 equals three from the left-hand side, a limit does exist. As we approach 𝑥 equals three from the right-hand side, a limit also exists. However, the limits are not the same. This means our function fails to meet the third condition and has a discontinuity. We have a point of discontinuity at 𝑥 equals three. And we need to decide what type of discontinuity it is.
A jump discontinuity occurs when the curve breaks at a particular place and starts somewhere else. This description applies to what is happening in our function. In a jump discontinuity, the limits from the left and from the right both exist. But they are not the same. But let’s go ahead and check the other three types.
An infinite discontinuity is a function that grows infinitely large as 𝑥 approaches 𝑎. This is another way of saying that the limit does not exist. Our limit is not approaching infinity. So we can cross that out. A removable discontinuity occurs when an otherwise continuous function has a hole in it. We call it removable because you can remove discontinuity by filling the hole. In a removable discontinuity, the limit as 𝑥 approaches 𝑎 from both sides is the same. This is not the case for our function.
And finally, we have an oscillating discontinuity. An oscillating discontinuity occurs when a function appears to be approaching two or more values simultaneously. An oscillating function might look something like this. And thus, it does not apply to our current function.
The type of discontinuity that occurs in this function at 𝑥 equals three is a jump discontinuity.
