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
Suddenly, the sun is eerily quiet: Where did the sunspots go?
'Moral victories' might spare you from losing again
Mammoth and mastodon behavior was less roam, more stay at home
snipez90
#2
Dec14-09, 01:33 PM
P: 1,105
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: 1,105
[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: 1,105
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: 1,105
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