Absolute value, continuous

I was given two questions, true and false, but am having trouble with them. They are...

"if f is continuous at a, so is absolute value of f."

and then
"if absolute value of f is continuous at a, so is f."

i am am at a loss at to how this works. The professor used a function for electric current as an example but It still seems to me that all points except zero would be continuous. Help!!!

Reply 1

ztibor

10

Original post by DottyPrawn

I was given two questions, true and false, but am having trouble with them. They are...

"if f is continuous at a, so is absolute value of f."

and then
"if absolute value of f is continuous at a, so is f."

i am am at a loss at to how this works. The professor used a function for electric current as an example but It still seems to me that all points except zero would be continuous. Help!!!

1. Yes

2, No

Reply 2

DottyPrawn

OP

Could you explain why? I feel like on the first one the continuous graph would still remain continuous if it was flipped over the x axis for negative numbers. On the second problem I think f(0) makes it false? I am very foggy on either problem.

Reply 3

Smaug123

15

Original post by DottyPrawn

Could you explain why? I feel like on the first one the continuous graph would still remain continuous if it was flipped over the x axis for negative numbers. On the second problem I think f(0) makes it false? I am very foggy on either problem.

First one: your intuition is sound. There's no way to become discontinuous without crossing the x-axis (as that's the only way to get the absolute function involved), and crossing the x-axis doesn't change continuity. Have you studied any analysis before? If so, you might well be able to come up with a proof yourself.

Second one: we want a counterexample which is as simple as possible. What's the simplest discontinuous function you know of? Can you make that into an example of a continuous-in-absolute-value discontinuous function? (Depending on whether your idea of "simplest" is the same as mine, of course, you might not be able to, but do tell us anyway.)

Reply 4

DottyPrawn

OP

Would f(x) = -2 for x<0 and 2 for x > or = to zero work?

So the absolute value would form a solid line at y=2 but f(0) would not be continuous because the limit from the negative would be -2 and the limit from the positive would be 2?

Reply 5

Noble.

17

Original post by DottyPrawn

Would f(x) = -2 for x<0 and 2 for x > or = to zero work?

So the absolute value would form a solid line at y=2 but f(0) would not be continuous because the limit from the negative would be -2 and the limit from the positive would be 2?

Yes, that works. For an even more striking example you can take Dirichlet's function

Unparseable latex formula:

f(x) = \begin{cases} 1, & \mbox{if } x\mbox{ is rational} \\ -1, & \mbox{if } x\mbox{ is irrational} \end{cases}

Then

|f(x)| = 1

for all

x \in \mathbb{R}

and so is continuous everywhere, yet

f(x)

is a nowhere continuous function (can you see why?)

Reply 6

DottyPrawn

OP

Would it be because without the absolute value there will always be going back and forth from 1 to -1 as you move across the x values?

(thanks so much for the help too!)

Reply 7

Smaug123

15

Original post by DottyPrawn

Would it be because without the absolute value there will always be going back and forth from 1 to -1 as you move across the x values?

(thanks so much for the help too!)

Yep, that's it - the irrationals and rationals are both dense in

\mathbb{R}

, so we can find an irrational and a rational arbitrarily close to any given point. Hence in any neighbourhood of a given point, the Dirichlet function takes both the values -1 and 1. (Alternatively, you could use the Intermediate Value Theorem: