finding continuous function $f$ such that $\int_0^1 x f(x) dx = 1$ and $\int_0^1 x^n f(x) dx = 0$ for $n\neq 1$

Question

Is it possible to find a continuous function $f(x)$ such that $$ \int_0^1 x f(x) dx = 1 $$ and $$ \int_0^1 x^n f(x) dx = 0 $$ where $n=0, 2, 3, 4, \dots$ I think this is not possible since $f(x)$ can be approximated by a polynomial, and if we take the second case, we get must be getting all coefficients zero for the approximating polynomial i.e. $f(x) = 0$, however, this contradictst he first case.

Answer 1

The answer is NO.$\int x^{n} (x^{2}f(x))dx=0$ for $n=0,1,2,\cdots$ and this implies $f\equiv 0$ (because $x^{2}f(x)=0$ for all $x$ and $f$ is continuous at $0$).