Beginner Functional Equation

Algebra Level 4

True or false :

If a function f ⁣ : R → R f \colon \mathbb R \to \mathbb R satisfies f ( x + y ) = f ( x ) + f ( y ) f(x+y)=f(x)+f(y) , then is always true that f ( x ) = a x f(x)=ax for some a ∈ R a\in\mathbb R .

True False

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展豪 張 by 展豪 張 , 22, Hong Kong

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1 solution

Vignesh S
Mar 28, 2016

For this to be possible f f must be continuous or monotonic. Since its not mentioned, nothing can be concluded.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes, but other than continuity, if it is given that the function is bounded below/above in some interval, or is increasing/decreasing in some interval, then also we can say f ( x ) = a x , ∀ x ∈ R f(x) = ax, \forall x \in \mathbb R .

Raushan Sharma - 5 years, 2 months ago

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