# The set of all continuous functions $f:[0,1]\rightarrow R$ satisfying $\int_{0}^{1}t^{n}f(t)dt=0$

I was thinking about the problem:

The set of all continuous functions $f:[0,1]\rightarrow R$ satisfying $$\int_0^1 t^n f(t) \, dt=0,\qquad n=1,2,\ldots$$

(a) $\text{is empty},$ (b) $\text{contains a single element},$ (c) $\text{is countably infinite},$ (d) $\text{is uncountably infinite}.$