Let $f_{n}$ be a sequence of continuous functions on the interval $[0,1]$ such that $f_{n}(x) \rightarrow f(x)$ for each $x$. For the three statements:

(a) $f_{n} \rightarrow f$ uniformly on $[0,1]$;

(b) $f$ is a continuous function;

(c) $\int_{0}^{1} f_{n}(x) d x \rightarrow \int_{0}^{1} f(x) d x$ as $n \rightarrow \infty ;$

say which of the six possible implications $(a) \Rightarrow(b),(a) \Rightarrow(c),(b) \Rightarrow(a),(b) \Rightarrow(c)$, $(c) \Rightarrow(a),(c) \Rightarrow(b)$ are true and which false, giving in each case a proof or counterexample.