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Please Help With New Problem due Wednsday

  1. Nov 19, 2007 #1
    Please Help With New Problem...due Wednsday!!!

    The question is the following..

    For all x f(x+1) = f(x)+1
    a. f(x) has to be unbroken
    b. f(x) is non-linear

    I just have to come up with ANY example of f(x) for that to be true. So basically come up with any function which is unbroken and is not linear so that f(x+1) = f(x)+1 is true.

    One example that he showed us in class was the step function, but he said we can't use that..so we have to make 1 up...can someone please HELP ME????



    I though i had it with f(x) = |x^2| , but i am not sure..

    f(x^2 + 1) = f(x^2) + 1??
     
    Last edited: Nov 19, 2007
  2. jcsd
  3. Nov 19, 2007 #2
    Do you mean f(x)=x^2? Because f(x^2) is not a "function", it's just what value to plug into some function 'f'. You need to find some f(x)="expression" such that f(x+1) = f(x)+1. So your idea, f(x^2), does not answer the question at hand. (And neither does f(x)=x^2.)

    Now, I'm unclear what is meant by "unbroken". Do you mean continuous?
     
  4. Nov 19, 2007 #3
    yes continuous and it has to be a NON-LINEAR function..and it has to follow the rule:

    f(x+1) = f(x) + 1
     
  5. Nov 19, 2007 #4
    anyone ....
     
  6. Nov 20, 2007 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Have you given any thought to what your f must be for x an integer?
    f(1) can be any number, of course, but then f(2)= f(1+1)= f(1)+ 1. f(3)= f(2+1)= f(1)+ 1+ 1= f(1)+ 2, f(4)= f(3+1)= f(3)+ 1= f(1)+2+1= f(1)+ 3. It's easy to see (or prove by induction) that as long as n is an integer, f(n)= f(1)+ n-1. The LINEAR function f(x)= C+ x-1 for C any constant would work fine but your example must not be a linear. I thought about using the "floor" function but that would not be continuous.

    I doubt that there is a continuous, non-linear, function satisfying that!
     
  7. Nov 20, 2007 #6

    My professor said that there are a lot of such functions...
     
  8. Nov 20, 2007 #7
    [tex]n*cos 2 \pi n[/tex]

    Would work for integer values. Maybe you heard him wrong and he wants something simple like this.
     
  9. Nov 20, 2007 #8

    I can't use any TRIGONOMETRY since we still didnt go over it in class, he wants a regular function.
     
  10. Nov 20, 2007 #9
    It's going to be quite the task to find a non-linear, non-periodic, continuous function satisfying [tex] f(x+1) = f(x)+1 [/tex]. You need to find more details about what this "professor" of yours is looking for, or at least make sure what you're telling us is accurate.

    Other than that tidbit, good luck on finding this needle in a haystack.
     
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