What if there are more solutions?

Algebra Level pending

Let f f be a continuous function f : R → R f:R\rightarrow R , that satisfies the following conditions for every x , y ∈ R x,y\in R :

  • f ( x + y ) = f ( x ) + f ( y ) f\left( x+y \right) =f\left( x \right) +f\left( y \right)
  • f ( 1 ) = 10 7 f\left( 1 \right) ={ 10 }^{ 7 }

Find the value of f ( e ) f\left( e \right) up to 2 decimal places.


The answer is 27182818.28.

Personal Data by Personal Data , 22, Norfolk Island

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1 solution

Tom Engelsman
Jul 10, 2020

This is a classical Cauchy functional equation that admits linear functions f ( x ) = A x + B f(x) = Ax + B as solutions. If we take x = y = 0 , x = y = 0, then we arrive at f ( 0 ) = 0 ⇒ B = 0 f(0)=0 \Rightarrow B = 0 as another boundary condition. Hence, f ( x ) = A x f(x) = Ax is the desired family of solutions. If f ( 1 ) = 1 0 7 , f(1) = 10^7, then f ( x ) = 1 0 7 x f(x) = 10^{7}x and f ( e ) = 27182818.28 . f(e) = \boxed{27182818.28}.

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