Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Can two different functions have an infinite number of solutions?

  1. Jun 17, 2013 #1
    Let f(x) and g(x) be non-piecewise defined functions that are defined for all real numbers. Furthermore, let f(x) and g(x) be continuous and differentiable at all points.
    Are there two functions f(x) and g(x) such that f(x)=g(x) for all points over some interval (a,b], and f(x)g(x) for all points over some interval (b,c)? Assume a≠b and b≠c.
    Basically, what I'm asking is this: can two different functions equal one another for all points over some interval? If I'm not making myself clear, see the attached picture:
    https://www.physicsforums.com/attachment.php?attachmentid=59652&stc=1&d=1371520767
    Thanks for your help. PS: This is my first post.
     

    Attached Files:

  2. jcsd
  3. Jun 17, 2013 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Non-piecewise defined isn't a very clearly defined term. For example is |x| allowed? What about f(x) = x if x > 0, and -x if x<0?
     
  4. Jun 17, 2013 #3

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Welcome to PF;
    ... yes they can. Unless you tighten your definition to the point where they can't.
     
  5. Jun 17, 2013 #4
    I realize that, but |x| isn't differentiable over its entire domain anyway.
     
  6. Jun 17, 2013 #5
    Could you give an example or two?
     
  7. Jun 17, 2013 #6

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Yeah but |x3| is.

    Whether it's differentiable or not is totally irrelevant anyway. Your question as posed doesn't make sense - saying "not piecewise defined" is not a well-defined statement.
     
  8. Jun 18, 2013 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Once you have said "differentiable for all x" you don't need "not piecewise defined".

    No, the fact that two functions are equal at every point on an interval does not mean they are equal for other points.
     
  9. Jun 18, 2013 #8
    Good point. I should redefine my terms.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Can two different functions have an infinite number of solutions?
Loading...