How can I prove $ f $ is a constant when $ \lim\limits_{h\rightarrow 0}\frac{1}{h^3}\int_{-h}^{h}f(x+t)t\,dt=0 $?

Question

Let $ f $ be a continuous function on $ (-\infty,\infty) $, such that for all $ x\in\mathbb{R} $, we have $ \lim\limits_{h\rightarrow 0}\frac{1}{h^3} \int_{-h}^h f(x+t)t\,dt=0 $. Please prove that $ f $ is a constant function. For this problem, here is my try. If I define $ F(t)=\int_0^t f(s) \, ds $, I have
\begin{eqnarray} \frac{1}{h^3}\int_{-h}^h f(x+t)t\,dt=\frac{1}{h^2}(F(x+h)+F(x-h))-\frac{1}{h^3} \int_{x-h}^{x+h}F(t) \, dt. \end{eqnarray} But I cannot solve it as the two terms above may not have limits. Can you give me some hints and references?