Under what conditions can you extend a function f(x) to be continuous at a point x=c?
Is this just rules for continuity or am I just really confused? If not what is this question asking?
Yes, rules of continuity. Has to have same limits from left and right, and has to be differentiable at that point.
Yes, use the definition of continuity. So, the limit of x to c has to exist. You can then define value of the function at x = c to be equal to that limit.
This question is asking about the conditions under which you can extend a function f(x) to be continuous at a specific point x=c. To understand this, it's important to have some background knowledge about continuity.
A function is said to be continuous at a point x=c if three conditions are satisfied:
1. The function f(x) is defined at x=c.
2. The limit of the function f(x) as x approaches c exists.
3. The value of the function f(x) at x=c is equal to the limit.
If any of these conditions are not met, the function is not continuous at x=c.
To extend a function f(x) to be continuous at x=c, you need to ensure that the above conditions are satisfied at that specific point. This means you may need to change or modify the function f(x) to meet these conditions.
To do this, you can follow these steps:
1. Ensure that the function f(x) is defined at x=c. Check if there are any gaps or points of discontinuity at x=c. If there are, you may need to redefine f(x) at that point.
2. Calculate the limit of the function f(x) as x approaches c. This involves evaluating the function as x gets as close as possible to c. If the limit does not exist, you may need to modify the function or apply appropriate techniques (such as L'Hospital's Rule).
3. Check if the value of the function f(x) at x=c is equal to the limit. If they are not equal, you may need to redefine the function or make specific adjustments to ensure that f(x) is continuous at x=c.
In summary, extending a function to be continuous at a specific point requires checking and ensuring that the function is defined, the limit exists, and the value of the function at that point matches the limit.