188 3 Real Analysis
Increase the left-hand side tox+
√
k; then square both sides. We obtainx^2 +k+ 2 x√
k≤k+kx^2 + 1 +x^2 ,which reduces to 0≤(x
√
k− 1 )^2 , and this is obvious. The induction is now com-
plete.
535.Find all functionsf:R→Rsatisfying
f(x^2 −y^2 )=(x−y)(f (x)+f (y)).536.Find all complex-valued functions of a complex variable satisfying
f(z)+zf ( 1 −z)= 1 +z, for allz.537.Find all functionsf:R{ 1 }→R, continuous at 0, that satisfy
f(x)=f(
x
1 −x)
, forx∈R\{ 1 }.538.Find all functionsf:R→Rthat satisfy the inequality
f(x+y)+f(y+z)+f(z+x)≥ 3 f(x+ 2 y+ 3 z)for allx, y, z∈R.539.Does there exist a functionf:R→Rsuch thatf(f(x))=x^2 −2 for all real
numbersx?
540.Find all functionsf:R→Rsatisfying
f(x+y)=f(x)f(y)−csinxsiny,for all real numbersxandy, wherecis a constant greater than 1.541.Letfandgbe real-valued functions defined for all real numbers and satisfying the
functional equation
f(x+y)+f(x−y)= 2 f (x)g(y)for allxandy. Prove that iff(x)is not identically zero, and if|f(x)|≤1 for all
x, then|g(y)|≤1 for ally.542.Find all continuous functionsf:R→Rthat satisfy the relation
3 f( 2 x+ 1 )=f(x)+ 5 x, for allx.