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Math Help - Find all functions

  1. #1
    Super Member dhiab's Avatar
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    Find all functions

    Find all functions f : IR ---> IR : ∀ (x,y) ∈ IRē :
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  2. #2
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    If we replace  f(t) ~~ by  \frac{1}{g(t)}

    Then it becomes

     \frac{1}{g(\frac{x+y}{2})}[ g(y) + g(x) ] = 2

     g(x) + g(y) = 2 g( \frac{x+y}{2})

    It implies that  g(t) = t   ~~ so  f(t) = \frac{1}{t}

    Is there another function satisifying the requirement ?
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  3. #3
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    How about f(t)=\frac1{at+b} for constants a, b not both zero?
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  4. #4
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    The equation g\left(\frac{x+y}2\right)=\frac{g(x)+g(y)}2 that occurs in simplependulum's reply is known as Jensen's equation.

    To solve it, put y=0 and then g\left(\frac x2\right)=\frac{g(x)+b}2 where b=g(0).

    Put x+y in this latter equation: g\left(\frac{x+y}2\right)=\frac{g(x+y)+b}2.

    And now we see that g(x+y)+b=g(x)+g(y).

    Finally, put h(x)=g(x)-b. Then h(x+y)=h(x)+h(y). This is the well-known Cauchy's equation.

    The most general continuous solution of Cauchy's equation is h(x)=ax where a is constant. Therefore g(x)=ax+b is the most general continuous solution of Jensen's equation.

    However, it must be said that there exist solutions of Cauchy's equation on \mathbb R which are not continuous, and they turn out to be extremely pathological. For example, if h is one of these solutions and I=(a,b) is any open interval then h(I) is dense in \mathbb R.

    So it makes sense in many cases to ask for conditions on the solution, such as continuity, or boundedness on a finite interval, or monotonicity, anything which would avoid these weird functions.
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