Show Continuity of f at x0 via Implied Convergence

  • Thread starter filter54321
  • Start date
  • Tags
    Continuity
In summary, the conversation discusses how to prove that a real valued function is continuous at a specific point x0 in its domain by using the limit definition of continuity and the epsilon-delta definition of the limit. It is shown that if for every sequence converging to x0, the limit of f(xn) is equal to f(x0), then f is continuous at x0. The conversation also addresses the question of how to move from the domain excluding x0 to the entire domain in order to satisfy the definition.
  • #1
filter54321
39
0

Homework Statement


Let f be a real valued function whose domain is a subset of R. Show that if, for every sequence xn in domain(f) \ {x0} that converges to x0, we have lim f(xn) = f(x0) then f is continuous at x0.


Homework Equations


Book definition of continuity:
"...f is CONTINUOUS at x0 in domain(f) if, for every sequence xn in domain(f) converging to x0, we have limnf(xn)=f(x0)..."


The Attempt at a Solution


The statement lim f(xn) = f(x0) would suggest that f(x0) exists, so leave that part of continuity aside for now.

What's the trick to get from domain(f) \ {x0} to domain(f) to satisfy the defintion?
 
Physics news on Phys.org
  • #2
Use the limit hypothesis to help show the standard epsilon-delta definition of continuity is satisfied at x0. You will use the epsilon-delta definition of the limit.
 

Related to Show Continuity of f at x0 via Implied Convergence

1. What is meant by "show continuity of f at x0 via implied convergence"?

This phrase refers to a method of proving that a function f is continuous at a specific point x0 by using the concept of implied convergence. This means that we can determine whether a function is continuous at a point by examining the behavior of the function's output as the input values approach the point in question.

2. How is implied convergence used to show continuity?

We can use the concept of implied convergence to show continuity by examining the limit of the function as the input values approach the point in question. If the limit exists and is equal to the function's value at the point, then we can conclude that the function is continuous at that point.

3. What is the difference between implied convergence and direct continuity?

Direct continuity refers to proving that a function is continuous at a point by directly examining the function's definition and showing that it meets the criteria for continuity. Implied convergence, on the other hand, uses the concept of limits to indirectly prove continuity by showing that the function's behavior approaches the point in question without necessarily evaluating the function at that point.

4. Can implied convergence be used to show discontinuity?

Yes, implied convergence can also be used to show that a function is discontinuous at a point. If the limit of the function at the point in question does not exist or is not equal to the function's value at that point, then we can conclude that the function is discontinuous at that point.

5. Why is implied convergence a useful method for proving continuity?

Implied convergence is a useful method for proving continuity because it allows us to prove continuity without having to evaluate the function at the point in question. This can be especially helpful when the function is complex or difficult to evaluate, making direct continuity proofs more challenging. Additionally, implied convergence can be used to prove continuity at points where the function is not defined, which is not possible with direct continuity proofs.

Similar threads

Monotonic sequence definition of Continuity of a function
    • Calculus and Beyond Homework Help
Replies
13
Views
986
Assume that f is a continuous, real-valued function
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Prove functions f and g are continuous in the reals
    • Calculus and Beyond Homework Help
Replies
23
Views
2K
Convergence of a continuous function related to a monotonic sequence
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Continuity of Function - f(x)=|cos(x)|
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Limit continuous function of rational numbers
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Prove limit comparison test for Integrals
    • Calculus and Beyond Homework Help
Replies
2
Views
867
Is there a smallest point in the interval [0,1] where f attains the value of 0?
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Showing that a pathological function is only continuous at 0
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Prove that the inner product converges
    • Calculus and Beyond Homework Help
Replies
15
Views
1K

Hot Threads

Recent Insights

Back
Top