How are jump discontinuities important when determining the area of a curve?

In summary: So, if you have a continuous function that is equivalent to a discontinuous function, then integration is still usually possible, but it will be more work.In summary, the professor explained how jump discontinuities must be taken into consideration when determining the area under the curve using the limit of #of rectangles definition. I did not really understand, or at least as well as I would have liked. Basically I want to know why a function can be integrated only on a continuous interval, and not on an interval with a discontinuity in it. Also, a function with a jump discontinuity is usually a piecewise defined function. Actually I think this isn't always so, as is the case with (x/|x|
  • #1
Terraist
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Hi

I'm a complete n00b at calculus (first day of the first year) so this question may appear a little...stupid. But I want to hammer into my head every little thing I hear in the lecture hall, so I might as well give it a go.

Today the professor explained how jump discontinuities must be taken into consideration when determining the area under the curve using the limit of #of rectangles definition. I did not really understand, or at least as well as I would have liked. Basically I want to know why a function can be integrated only on a continuous interval, and not on an interval with a discontinuity in it.

Also, a function with a jump discontinuity is usually a piecewise defined function. Actually I think this isn't always so, as is the case with (x/|x|). Is it true that a piecewise defined function can't be integrated in one step, and each part has to be analyzed separately?
 
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  • #2
If g is continuous on [a,b], f has only finitely many jump discontinuities on [a,b], and f(x)=g(x) for all x where f is continuous, then the integral of g from a to b is equal to the integral of f from a to b. Or ...

[tex]\int_a^bg = \int_a^bf[/tex]
 
  • #3
I'm sorry, I don't really follow. Are you saying that if I find a continuous function equivalent to the continuous parts of the jump function, and integrate it on [a,b], it would equal the integral of the jump function on that same interval? Could you perhaps provide a pictorial representation?
 
  • #4
Sorry! I got my terms mixed up! I meant removable discontinuity in my post. However, there is a similar result for jump discontinuities. I'll work on posting it in the next few hours when I find the time. Again, sorry for the confusion.
 
  • #5
The best thing to do with jump discontinuities is to break the problem into "pieces" with the boundary at the points of discontinuity. Then you have several "regular" areas and the entire area is the sum of those.

First, I would say that your example of |x|/x is not an example of a jump discontinuity since the value is not defined at x= 0. If you were, for example to define f(x)= x/|x| for x not 0, f(x)= a for x= 0, then there is, indeed a jump discontinuity at x= 0. But now the function can be written as a piecwise continuous function- f(x)= 1 for x> 0, f(0)= a, f(x)= -1 for x< 0.

Also, I would not say that can't be integrated directly, only that it is much easier, using the usual techniques of integration, to divide it into "continuous parts". And I suspect that was what your teacher was trying to say. Integration is done pretty much by "remembering" anti-derivatives- and we typically learn continuous functions because those are easiest.
 

1. What is a jump discontinuity?

A jump discontinuity is a type of discontinuity in a function where there is a sudden change or "jump" in the value of the function at a specific point. This means that the function is not continuous at that point, and there is a break or gap in the graph of the function.

2. How does a jump discontinuity affect the area under a curve?

Jump discontinuities can significantly affect the calculation of the area under a curve because they create gaps or breaks in the graph of the function. This means that the area calculation will be interrupted at the point of the jump, and the resulting area will not accurately represent the total area under the curve.

3. Can a jump discontinuity be ignored when calculating the area under a curve?

No, a jump discontinuity cannot be ignored when calculating the area under a curve. It is an essential factor to consider because it can significantly affect the accuracy of the area calculation. Ignoring a jump discontinuity can lead to incorrect results and a distorted representation of the curve.

4. How can jump discontinuities be identified when determining the area of a curve?

Jump discontinuities can be identified by visually inspecting the graph of the function. They appear as sudden changes or "jumps" in the graph, where the function value changes abruptly. In some cases, a jump discontinuity can also be identified by analyzing the function's equation and determining if there are any discontinuities at specific points.

5. Are there any special methods for handling jump discontinuities when calculating the area under a curve?

Yes, there are special methods for handling jump discontinuities when calculating the area under a curve. One method is to split the function into smaller intervals, each containing only one jump discontinuity. The area under each interval can then be calculated separately and added together to get the total area. Another method is to use the Riemann sum approach, where the jump discontinuity is treated as a separate point and the area is calculated using the left and right-hand limits at that point.

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