Register to reply

[f(x)'^n continuous at a point (EXAM IN 4 hrs.)

by llursweetiell
Tags: continuity
Share this thread:
llursweetiell
#1
Dec14-09, 12:25 PM
P: 11
1. The problem statement, all variables and given/known data
Suppose f: D-->R is continuous at a. Let n >1 be a positive integer. using the epsilon-delta definition of continuity, prove g(x)=[f(x)]^n is continuous as a


2. Relevant equations
i know how to do it as a sequence proof; but i don't know how to use the epislon/delta definition to prove it.


3. The attempt at a solution
i tried using the composition proof too, but that didn't work. any help?
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
snipez90
#2
Dec14-09, 01:33 PM
P: 1,104
Um, I'm assuming you mean g = f nested n times since you mentioned the "composition proof". If this is the case, I don't even think the theorem is true since if n = 2, you would need f to be continuous at f(a), but f(a) might not even be in D.
llursweetiell
#3
Dec14-09, 01:47 PM
P: 11
the problem states:
suppose f:D--> R is continuous at a. Let n >1 be a positive integer. Using the epsilon and delta definition of continuity, prove g(x)=[f(x)]^n is continuous at a.

Does that help?

Also, the composition way did not work. which is why i'm unsure of how to go about tackling the problem.

snipez90
#4
Dec14-09, 02:04 PM
P: 1,104
[f(x)'^n continuous at a point (EXAM IN 4 hrs.)

Sorry, the comment about the composition of functions threw me off. I should have known that the solvable interpretation is the right one.

Anyways, you need to estimate |[f(x)]^n - [f(a)]^n| knowing that you can bind |f(x) - f(a)| by any positive number. But a^n - b^n has a very nice and symmetric formal factorization, namely

[tex]a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + ... + b^{n-2}a + b^{n-1})[/tex]

Thus you can replace a with f(x) and b with f(a) to get |[f(x)]^n - [f(a)]^n| equal to |f(x) - f(a)| times a bunch of terms within an absolute value (this is key) as a result of the factorization above. But now can you see what to bind |f(x) - f(a)| by to ensure that |[f(x)]^n - [f(a)]^n| is less than say, epsilon?
llursweetiell
#5
Dec14-09, 06:51 PM
P: 11
So i wrote,
We know that f is continuous at a. to prove that g is continuous at a:
let epsilon be greater than 0 and let delta > 0 such that d=epsilon/[tex](f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})[/tex]

then,
for |x-a| < delta, then |f(x)^n-f(a)^n|=[tex]f(x)^n - f(a)^n = (f(x)-f(a))(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})[/tex]
< (f(x)-f(a)) * delta = epsilon. So g is continuous at a.

Is that what you meant?
snipez90
#6
Dec14-09, 11:02 PM
P: 1,104
You have the right idea, but you don't actually get to choose delta here. Since f is continuous at a, there is already some delta for which |x-a| < delta implies
[tex]|f(x) - f(a)| < \frac{\varepsilon}{(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})}.[/tex]

But this is exactly what we need.
llursweetiell
#7
Dec14-09, 11:37 PM
P: 11
[QUOTE=
[tex]|f(x) - f(a)| < \frac{\varepsilon}{(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})}.[/tex]

QUOTE]

do you mean f(x)^n-f(a)^n < all of the factorization?
snipez90
#8
Dec15-09, 12:46 AM
P: 1,104
Eh, perhaps it would be clearer if I just pieced together the proof. Let epsilon > 0 be given. By the continuity of f at a, there exists some delta > 0 such that |x-a| < delta implies
[tex]
|f(x) - f(a)| < \frac{\varepsilon}{|f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1}|}.
[/tex]
But then
[tex]
|f(x)^n - f(a)^n| = |f(x)-f(a)||f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1}| < \varepsilon.
[/tex]
llursweetiell
#9
Dec15-09, 09:46 AM
P: 11
oh that makes sense. thank you so much! =)


Register to reply

Related Discussions
Prove function continuous at only one point Calculus & Beyond Homework 1
Is the anti-derivative of a continuous function continuous? Calculus 13
How to show continuous at each point in R^2 Calculus & Beyond Homework 4
Is a graph Continuous and differentiable at a given point Calculus & Beyond Homework 13
Fixed Point in Continuous Set Calculus & Beyond Homework 5