# Real Analysis: Epsilon- Delta HELP NEEDED

• May 6th 2008, 08:14 PM
Vitava61
Real Analysis: Epsilon- Delta HELP NEEDED
I have a final on Friday and i'm still not sure about a lot of things for this class... Here's this question:

let f: D-->R and define I f I: D--> R by I f I (x) = I f(x) I

a) suppose that f: D-->R is continuous at cED. Prove using an epsilon-delta argument that I fI is continuous at c.
b) Prove that I f I is continuous at c it does not necessarily follow that f is continuous at c.

I am completely lost on this problem, my professor is horrible

the I f I like represents absolute value? that's what all the I's stand for in the problem. Anyone know how to start this problem?
• May 6th 2008, 09:14 PM
TheEmptySet
Quote:

Originally Posted by Vitava61
I have a final on Friday and i'm still not sure about a lot of things for this class... Here's this question:

let f: D-->R and define I f I: D--> R by I f I (x) = I f(x) I

a) suppose that f: D-->R is continuous at cED. Prove using an epsilon-delta argument that I fI is continuous at c.
b) Prove that I f I is continuous at c it does not necessarily follow that f is continuous at c.

I am completely lost on this problem, my professor is horrible

the I f I like represents absolute value? that's what all the I's stand for in the problem. Anyone know how to start this problem?

for part a)

let $\epsilon>0$ be given.
Since f is continous at c choose a delta

$|x-c|<\delta, |f(x)-f(c)|< \epsilon$

We wish to show that $||f(x)|-|f(c)||<\epsilon$

by the triangle inequality||a|-|b||<|a-b|

$||f(x)|-|f(c)||<|f(x)-f(c)|<\epsilon$

QED.

I will try to think of a counter example for b.
• May 6th 2008, 10:06 PM
Jhevon
Quote:

Originally Posted by Vitava61
b) Prove that I f I is continuous at c it does not necessarily follow that f is continuous at c.

The |'s mean absolute values

example: $f(x) = \left \{ \begin{array}{rlr} 1 & \mbox{ if } x \ge 0 & \\ & & c = 0 \\ -1 & \mbox{ if } x < 0 & \end{array} \right.$
• May 7th 2008, 06:55 PM
Vitava61
Can you explain to me how that counter example proves that is I f I is continuous at c, it does not necessarily follow that f is continuous at c?

Thanks for all your help, I've been able to either do, or make sense out of everything else.
• May 7th 2008, 06:58 PM
Jhevon
Quote:

Originally Posted by Vitava61
Can you explain to me how that counter example proves that is I f I is continuous at c, it does not necessarily follow that f is continuous at c?

Thanks for all your help, I've been able to either do, or make sense out of everything else.

the function i gave is not continuous at x = 0. As i hope is obvious.

however, if you take the absolute value of the function, we have |f(x)| = 1. this is a polynomial and is continuous everywhere

thus |f(x)| is continuous, but f(x) isn't