Let$f:R\mapsto R$ be a continuous function such that for all $x\in R$, we have,
$\int_{0}^{1}f(xt)dx=0$ (*)then,
A) There is an $f$ satisfying (*) that takes both positive and negative values.
B) There is an $f$ satisfying (*) that is zero at infinitely many points, but not identically zero
C) $f$ must be identically zero on the whole of R.
D) There is an $f$ satisfying (*) that is identically zero on $(0,1)$ but not identically zero on the whole $R$
My attempt:
Consider,
$\int_{0}^{1}f(xt)dx=0$
Substituting $xt$=$y$ we get
$\int_{0}^{t} \frac{f(y)}{t}dy=0$
Differentiating both the sides w.r.t to $t$ ,
$f(t)=0$
$\implies$ there exists $f$ that is identically $0$? Is this even correct?
