# Thread: Real Analysis: Epsilon- Delta HELP NEEDED

1. ## 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?

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

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

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

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

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

QED.

I will try to think of a counter example for b.

3. 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: $\displaystyle f(x) = \left \{ \begin{array}{rlr} 1 & \mbox{ if } x \ge 0 & \\ & & c = 0 \\ -1 & \mbox{ if } x < 0 & \end{array} \right.$

4. 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.

5. 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