When one-sided limits are not equal to each other at a point, what can be said about the continuity of the function at that point?
I know that when one-sided are not equal to each other then the limit does not exist at that point. but i do not know what can be said about the continuity of the function at that point
the discontinuity is non-removable.
There's no way to define f(x) there so that the two limits are the same and equal f(x).
Well, if the one-sided limits at a point are not equal, it indicates a discontinuity in the function at that point. In other words, the function would not be considered continuous at that particular point. It's like having two sides of an argument that just can't get along. They're not able to meet in the middle and have a nice, smooth transition. So, in short, if the one-sided limits are not equal at a point, it means the function is not continuous at that point.
When the one-sided limits are not equal to each other at a point, the function is not continuous at that point. Continuity of a function requires that the limit exists at that point, and in the case where the one-sided limits are not equal, the limit does not exist. Thus, the function is not continuous at that point.
When the one-sided limits of a function at a point are not equal to each other, it implies that the function is not continuous at that point. Let's break down why this is the case:
To determine the continuity of a function at a certain point, we need three conditions to be satisfied:
1. The function must be defined at the point in question.
2. The limit of the function as it approaches the point from both sides must exist.
3. The limit must be equal to the value of the function at that point.
So, if the one-sided limits are not equal at a point, it means that the second condition is not satisfied. In other words, the function does not have a well-defined limit at that point.
This lack of a defined limit implies that the function has some kind of discontinuity at that point. There are different types of discontinuities, such as removable (jump) discontinuities, infinite discontinuities, or oscillating (essential) discontinuities. The specific type of discontinuity can be determined by further analyzing the behavior of the function near that point.
In summary, when the one-sided limits are not equal to each other at a point, we can conclude that the function is not continuous at that point.