A function can be continuous at every point of a deleted neighborhood of some point and still not have a limit at the point which is deleted.
I'm not even too sure what this means in calculus terms. Can you please explain it better? Thank you.
Consider f(x) = 1/x
f is continuous at every point in every neighborhood not including x=0.
Yet f does not have a limit at x=0; in fact it is not even defined at x=0.
Certainly! In calculus, we often study the concept of continuity and limits.
When we say a function is continuous at a point, it means that the function is defined at that point, and the limit of the function exists and matches the value of the function at that point. Essentially, it implies that there are no abrupt jumps or breaks in the graph of the function at that particular point.
A deleted neighborhood of a point is an interval that contains all the points near the given point, except the point itself. For example, the deleted neighborhood of a point x = 2 would be the interval (1.9, 2.1), which includes all values from 1.9 to 2.1, excluding x = 2.
Now, let's consider the statement: "A function can be continuous at every point of a deleted neighborhood of some point and still not have a limit at the point which is deleted."
This means that a function can exhibit continuity at every point within a deleted neighborhood of a specific point. In other words, the function appears continuous when you zoom in on that particular point. However, when you consider the point itself (which is deleted from the neighborhood), the function may not have a limit at that point.
To visualize this concept, imagine a function that oscillates rapidly between two values within the deleted neighborhood but behaves differently when you consider the point itself. For example, consider the function f(x) = sin(1/x). This function oscillates infinitely as x approaches 0 but does not have a limit at x = 0. However, if you look at any finite deleted neighborhood around x = 0, the function appears to be continuous.
So, the statement reminds us that having continuity at every point within a deleted neighborhood does not guarantee a limit at that deleted point. It highlights the distinction between local behavior (within the deleted neighborhood) and the behavior at the specific point that is excluded from the neighborhood.
I hope this explanation clarifies the concept for you!