Continuous real function/Approximation properties/Fact/Proof/Exercise

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function. Show the following statements.

1. The function ${}f$ is uniquely determined by its values on ${}\mathbb {Q}$.
2. The value ${}f(a)$ is determined by the values ${}f(x)$, ${}x\neq a$.
3. If for all ${}x<a$, the estimate

   ${}f(x)\leq c\,$

   holds, then also

   ${}f(a)\leq c\,$

   holds.

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