MathDB

Let

\mathbb{R}

be the set of real numbers. Let

f:\mathbb{R}\rightarrow\mathbb{R}

be a function such that

f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2

for every

x,y\in\mathbb{R}

. Assume that the inequality is strict for some

x_0,y_0\in\mathbb{R}

.
Prove that either

f(x)\geqslant 0

for every

x\in\mathbb{R}

f(x)\leqslant 0

for every

x\in\mathbb{R}

A2