Question:
f is continuous on $\mathbb{R},f(0)=0$,$\forall x\in\mathbb{R},\lim\limits_{\delta \to 0}\frac{1}{\delta^3}\int_{-\delta}^{\delta} f(x+t)t dt=0$.
$\forall\epsilon>0$,find a polynomial P such that $|P−f|<\epsilon |x|,\forall x\in [-M,M]$.
Attempt:
If we assume that $f$ is differentiable at $0$,then define $g=f(x)/x(x\neq 0),f'(0)(x=0)$,$g$ is continuous on $\mathbb{R}$,thus $\exists Q_N:|Q_N-g|<\epsilon\implies |Q_Nx-f|<\epsilon|x|$.
Original Problem:
f is continuous on $\mathbb{R}$,$\forall x\in\mathbb{R},\lim\limits_{\delta \to 0}\frac{1}{\delta^3}\int_{-\delta}^{\delta} f(x+t)t dt=0$.Prove that $f$ is constant.
I've tried to use the Weierstrass theorem and use polynomials to substitute $f$,where the question arose.If we can prove it,then it's not hard to prove the original problem by caculating integrals of polynomials.
Anything about the question or the original problem is appreciated!
