Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    1
    ReputationRep:
    Name:  continuity.png
Views: 89
Size:  84.3 KB

    I'm stuck with part b) but I'm not sure if I've done a) correctly neither.

    For a) I took the left and right hand limits 'a-' and 'a+' and deduced that both ----> 0 and so the limit of 'a' must also go to 0, suggesting the function is continuous at x=a and I did this for 'b-' and 'b+' too also meaning that the function must be continuous at x=b too! Is this the correct approach?
    • Study Helper
    Online

    13
    Study Helper
    (Original post by As_Dust_Dances_)
    Name:  continuity.png
Views: 89
Size:  84.3 KB

    I'm stuck with part b) but I'm not sure if I've done a) correctly neither.

    For a) I took the left and right hand limits 'a-' and 'a+' and deduced that both ----> 0 and so the limit of 'a' must also go to 0, suggesting the function is continuous at x=a and I did this for 'b-' and 'b+' too also meaning that the function must be continuous at x=b too! Is this the correct approach?
    The working for the limits "a-" and "b+" are outside the domain of f, and don't come into it.

    Although I'm not an analysis expert, I'd say (x-a) is continuous on [a,b], as is (b-x), therefore their product is also. Square rooting is a continuous function (perhaps iffy in an analysis course), and (x-a)(x-b) is in the domain of that function, hence the whole thing is continuous on [a,b].

    For the second one f is not defined at x=b. Aside from that proceed as per first one.

    I'm sure someone more on the ball will correct me if I'm in error here.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by ghostwalker)
    The working for the limits "a-" and "b+" are outside the domain of f, and don't come into it.

    Although I'm not an analysis expert, I'd say (x-a) is continuous on [a,b], as is (b-x), therefore their product is also. Square rooting is a continuous function (perhaps iffy in an analysis course), and (x-a)(x-b) is in the domain of that function, hence the whole thing is continuous on [a,b].

    For the second one f is not defined at x=b. Aside from that proceed as per first one.

    I'm sure someone more on the ball will correct me if I'm in error here.
    I was actually thinking along the lines of the "product" such as splitting it up into f(x) and g(x) then using the lim x --> a, but I'm not certain. I'm planning on seeing my lecturer tomorrow about it, I think I understand the basics of continuity, it's just questions like these that throw me off slightly and whether or not my lecturer expects me to do it some specific way I'm not sure. Thanks for your help anyway! (Won't let me give you rep)
    Offline

    17
    ReputationRep:
    It's a little hard without context (i.e. some idea of how long you're supposed to spend on the question).

    For a bit of "light discussion", what ghostwalker says is fine. But if the exam paper implies it's, say, 20 minutes work, then I would think a proper epsilon-delta discussion would be more appropriate.
    • Study Helper
    Online

    13
    Study Helper
    The other thought that struck me is uniform continuity, or not, as the case may be.
    Offline

    17
    ReputationRep:
    (Original post by ghostwalker)
    The other thought that struck me is uniform continuity, or not, as the case may be.
    I think that's unlikely; a continuous function on a closed interval is always uniformly continuous.
    • Study Helper
    Online

    13
    Study Helper
    (Original post by DFranklin)
    a continuous function on a closed interval is always uniformly continuous.
    Granted, but wadr the second function is only defined on [a,b)
 
 
 
  • 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
    Would you rather give up salt or pepper?
    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

    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
  • 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

    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.