What qualifies as an infinite discontinuity?

[shinwolf14]

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.

 

[Mark44]

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.

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."

 

[shinwolf14]

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