You are Here: Home >< Maths

# Absolute value, continuous Watch

1. 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!!!
2. (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
3. 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.
4. (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.)
5. 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?
6. (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

Then for all and so is continuous everywhere, yet is a nowhere continuous function (can you see why?)
7. 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!)
8. (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 , 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: while , so if f were continuous, f would have a root between e and 3. But f is nowhere 0 - it is only ever 1 or -1. Hence f is not continuous.)

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: July 2, 2014
Today on TSR

### Oxford interview invitations

When can you expect yours?

### Official Cambridge interview invite list

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.