What qualifies as an infinite discontinuity?

In summary, an infinite discontinuity exists when at least one of the one-sided limits of a function is infinite. This can occur when one limit approaches infinity and the other is finite, or when both limits approach infinity. It is different from a jump discontinuity, where both one-sided limits are finite. This type of discontinuity is also known as an essential discontinuity, and it is important to distinguish it from a removable discontinuity.
  • #1
shinwolf14
14
0
Hello everyone. I am currently having trouble actually defining what qualifies as an infinite discontinuity. I have read several sources that state that both of the one sided limits must approach infinity (positive, negative or both). My problem is what happens when only one of the one sided limits approach infinity and the other is finite. This is not a jump discontinuity because both of the one sided limits are not finite. According to the definition that I have read, it doesn't qualify as an infinite discontinuity either. I have crudely drawn what I am trying to describe. I am pretty sure this still is classified as an infinite discontinuity but I just wanted to be sure.

Ek28693.png
 
Physics news on Phys.org
  • #2
shinwolf14 said:
Hello everyone. I am currently having trouble actually defining what qualifies as an infinite discontinuity. I have read several sources that state that both of the one sided limits must approach infinity (positive, negative or both). My problem is what happens when only one of the one sided limits approach infinity and the other is finite. This is not a jump discontinuity because both of the one sided limits are not finite. According to the definition that I have read, it doesn't qualify as an infinite discontinuity either. I have crudely drawn what I am trying to describe. I am pretty sure this still is classified as an infinite discontinuity but I just wanted to be sure.

Ek28693.png
I would call it an infinite discontinuity. A simpler example than your graph shows is
##f(x) = \begin{cases} 1 & \text{if } x \le 0 \\ \frac 1 x & \text{if } x > 0\end{cases}##
In my example, ##\lim_{x \to 0^-} f(x) = 1##, but ##\lim_{x \to 0^+} f(x) = \infty##

Another site I found would agree with me -- http://www.milefoot.com/math/calculus/limits/Continuity06.htm
An infinite discontinuity exists when one of the one-sided limits of the function is infinite.
I believe that their definition would also include the case where both one-side limits are infinite. If so, their definition would be clearer if it said "when at least one of the one-side limits of the function is infinite."
 
  • #3
Mark44 said:
If so, their definition would be clearer if it said "when at least one of the one-side limits of the function is infinite."

This is what I assumed. When I learned Calculus 1, we classified this as an essential discontinuity, compared to the removable discontinuity. Obviously we went over infinite discontinuities but they typically were asymptotic
 

1. What is an infinite discontinuity?

An infinite discontinuity is a type of discontinuity in a function where the limit of the function at a certain point does not exist because it approaches positive or negative infinity.

2. How is an infinite discontinuity different from other types of discontinuities?

An infinite discontinuity is different from other types of discontinuities, such as jump or removable discontinuities, because the limit of the function at the point of discontinuity is undefined or infinite.

3. What causes an infinite discontinuity?

An infinite discontinuity can be caused by a vertical asymptote in the function, where the function values approach positive or negative infinity as the x-values approach a certain point.

4. How can we identify an infinite discontinuity?

An infinite discontinuity can be identified by graphing the function and observing a vertical asymptote at the point of discontinuity, or by calculating the limit of the function at the point of discontinuity and determining if it is undefined or infinite.

5. Can an infinite discontinuity be removed?

No, an infinite discontinuity cannot be removed or "fixed" like a removable discontinuity. It is a fundamental property of the function and cannot be altered without changing the entire function.

Similar threads

Replies
1
Views
652
Replies
10
Views
3K
  • Quantum Physics
Replies
5
Views
834
Replies
1
Views
1K
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
858
  • Topology and Analysis
Replies
10
Views
3K
  • General Math
Replies
4
Views
4K
Replies
4
Views
2K
Back
Top