Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    Hi,

    I am stuck with this past paper question:

    give the definition of a function f being continuous at x0, which I have done

    then it says for the following three statements say if it is true or false and provide a counter example or proof:

    1. if f and g are continuous at x0 then so is f + g

    2. if f + g is continuous at x0 then so are f and g

    3. if f and f + g are continuous at x0 then so is g

    I can do 2. as if you let f(x) = 1 / x and g(x) = 1 - f(x), then f is discontinuous and f + g is continuous but im not sure about the other 2?

    any one help?

    thanks
    Offline

    14
    ReputationRep:
    1 is true - you may be able to do this with an epsilon/2 argument but I'm not certain. It is definitely true though.
    • PS Helper
    Offline

    14
    PS Helper
    For #1 you can use the triangle inequality. That is, |(f+g)(x_0)| \le |f(x_0)| + |g(x_0)| (think how to apply this here). For #3 you can do a similar thing, noting that (f+g) - f = g.
    Offline

    18
    ReputationRep:
    (Original post by Big_Sam)
    I can do 2. as if you let f(x) = 1 / x and g(x) = 1 - f(x), then f is discontinuous and f + g is continuous but im not sure about the other 2?
    This isn't actually valid: f and g aren't defined at x = 0, (and therefore neither is f+g).
    Offline

    16
    ReputationRep:
    (Original post by DFranklin)
    This isn't actually valid: f and g aren't defined at x = 0, (and therefore neither is f+g).
    easily remedied by saying that f and g are both =1/2 at x=0.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: May 20, 2010

University open days

  • University of East Anglia
    All Departments Open 13:00-17:00. Find out more about our diverse range of subject areas and career progression in the Arts & Humanities, Social Sciences, Medicine & Health Sciences, and the Sciences. Postgraduate
    Wed, 30 Jan '19
  • Solent University
    Careers in maritime Undergraduate
    Sat, 2 Feb '19
  • Sheffield Hallam University
    City and Collegiate Campus Undergraduate
    Sun, 3 Feb '19
Poll
The new Gillette ad. Is it:
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

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

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