A function f : R → R where R is a set of real numbers satisfies the equation f ( 3 x + y ) = 3 f ( x ) + f ( y ) + f ( 0 ) for all x , y ∈ R . If the function is differentiable at x = 0 then show

Question

A function f : R → R where R is a set of real numbers satisfies the equation f ( 3 x + y ​ ) = 3 f ( x ) + f ( y ) + f ( 0 ) ​ for all x , y ∈ R . If the function is differentiable at x = 0 then show that it is differentiable for all x in R

TestHub · Accepted Answer

f ( 3 x + y ​ ) − 3 f ( x ) + f ( y ) + f ( 0 ) ​ . ​ h → 0 lim ​ h f ( 0 + h ) − f ( 0 ) ​ = exist h → 0 lim ​ h f ( x − h ) − f ( x ) ​ = h → 0 lim ​ h f ( 3 3 x + 3 h ​ ) − f ( 3 3 x + 0 ​ ) ​ h → 0 lim ​ h 1 ​ [ 3 f ( 3 x ) + f ( 3 h ) + f ( 0 ) ​ − 3 f ( 3 x ) + f ( 0 ) + f ( 0 ) ​ ] = h → 0 lim ​ h 1 ​ [ 3 f ( 3 h ) − f ( 0 ) ​ ] = h → 0 lim ​ 3 h f ( 3 h ) − f ( 0 ) ​ = f ′ ( 0 ) ​

Mathematics - Continuity - Differentiability Question with Solution | TestHub

Doubts & Discussion