Calculation of $\int_a^b f(x)\,dx$

Question

Let $f:\Bbb R\to \Bbb R$ a continous function and $a,b\in \Bbb R,a<b$. Calculate $\int_a^b f(x)\,dx$ if $f(tx)\ge f(x)$ for any $t>0$ and $x\in \Bbb R$

I obtained just that $f(x)\ge f(0)$ but I do not think it is helpful.

Answer 1

Put $t=\frac y x$ to get $f(y) \geq f(x)$ for all $x,y>0$. Can you show now that $f$ is a constant on $(0,\infty)$? A similar argument would show that $f$ is a constant on $(-\infty,0)$. By continuity, $f$ is a constant.

Answer 2

Take $x,y \in \mathbb{R^+}$. You have that $f(x)=f(\frac{x}{y}y) \geq f(y)$ and $f(y)=f(\frac{y}{x}x) \geq f(x)$. So $f$ is constant on $\mathbb{R^+}$. In the same way you obtain that $f$ is constant on $\mathbb{R^-}$. Since $f$ is continuos $f(x)=k \;\; \forall x \in \mathbb{R}$. So you have that $\int_a^bf(x) dx=k(b-a)$