Continuity of one function, implies continuity of another?

In summary, the conversation discusses the continuity of integrable functions and their integrals. While the continuity of f is not necessary for its integral to be continuous, the continuity of the integral does not imply the continuity of f.
  • #1
bobby2k
127
2
Hi

Lets say that f(x) is continuous. Then [itex] \int_0^x \! f(t)dt=G(x)[/itex] is continuous. (I don't think you have to say that f need to be continuous for this, all we need to say is that f is integrable?, or do we need continuity of f here?)

But my main question is about the converse. let's say that [itex] \int_0^x \! f(t)dt=G(x)[/itex] is continuous, does that imply that f is continuous?

Have a nice sunday.
 
Last edited:
Physics news on Phys.org
  • #2
bobby2k said:
Hi

Lets say that f(x) is continuous. Then [itex] \int_0^x \! f(t)dt=G(x)[/itex] is continuous. (I don't think you have to say that f need to be continuous for this, all we need to say is that f is integrable?, or do we need continuity of f here?)

This is true for any integrable function.

But my main question is about the converse. let's say that [itex] \int_0^x \! f(t)dt=G(x)[/itex] is continuous, does that imply that f is continuous?

No: by the above, [itex]f[/itex] does not need to be continuous for its integral to be continuous.
 
  • Like
Likes 1 person
  • #3
pasmith said:
This is true for any integrable function.



No: by the above, [itex]f[/itex] does not need to be continuous for its integral to be continuous.

Hehe, ofcourse, thanks.
 

1. What is the definition of continuity for a function?

The definition of continuity for a function is that the function must have a defined value at each point on its domain, and the limit of the function at each point must be equal to the value of the function at that point.

2. How does the continuity of one function imply the continuity of another?

If two functions are defined on the same domain and the first function is continuous, then the second function will also be continuous on that domain if it is a composition, sum, or product of the first function and other continuous functions.

3. Can a discontinuous function imply the continuity of another function?

No, a discontinuous function cannot imply the continuity of another function. In order for a function to be continuous, it must have a defined value at each point on its domain and its limit must exist at each point. If a function is discontinuous, one or both of these criteria are not met, and the continuity of another function cannot be implied.

4. How can you prove that the continuity of one function implies the continuity of another?

To prove that the continuity of one function implies the continuity of another, you can use the definition of continuity and show that the limit of the second function at each point is equal to the value of the function at that point. You can also use the continuity rules for compositions, sums, and products to show that the second function is continuous if it is a composition, sum, or product of the first function and other continuous functions.

5. Are there any exceptions to the rule that the continuity of one function implies the continuity of another?

Yes, there are some exceptions to this rule. For example, if the second function is a quotient of the first function and another function, the continuity of the second function may not be implied by the continuity of the first function. In this case, the other function must also be continuous for the continuity of the second function to be implied.

Similar threads

I On convolution theorem of Laplace transform: Schiff
    • Calculus
Replies
4
Views
723
I Lipschitz continuity of vector-valued function
    • Calculus
Replies
3
Views
1K
A Summing over continuum and uncountable numerocities
    • Calculus
Replies
1
Views
907
MHB Differentiability and continuity
    • Calculus
Replies
1
Views
954
I Second derivative of chained function
    • Calculus
Replies
14
Views
1K
I Multivariable fundamental calculus theorem in Wald
    • Calculus
Replies
2
Views
1K
I Some questions on l'Hôpital rules
    • Calculus
Replies
9
Views
2K
I Substitution in a definite integral
    • Calculus
Replies
6
Views
1K
I Looking for Guarantees that the method of fixed-point iteration will work
    • Calculus
Replies
22
Views
1K
Insights Can You Solve the Mathematical Mystery Behind the Art of Integration?
    • Calculus
Replies
10
Views
3K

Hot Threads

Recent Insights

Back
Top