Let $a$ be a given real number in $[0,1] .$ Prove that there exists a unique $g \in C[0,1]$ such that
$$
f(a)=\int_{0}^{1} f(x) g(x) d x \text { for all } f \in C[0,1]
$$
$$
f(a)=\int_{0}^{1} f(x) g(x) d x \text { for all } f \in C[0,1]
$$
