functional equations - Real Analysis Proofs: Additive Functions

I'm new here and could really use some help please:

Let $f$ be an additive function. So for all $x,y \in \mathbb{R}$, $f(x+y) = f(x)+f(y)$.

1. Prove that if there are $M>0$ and $a>0$ such that if $x \in [-a,a]$, then $|f(x)|\leq M$, then $f$ has a limit at every $x\in \mathbb{R}$ and $\lim_{t\rightarrow x} f(t) = f(x)$.




2. Prove that if $f$ has a limit at each $x\in \mathbb{R}$, then there are $M>0$ and $a>0$ such that if $x\in [-a,a]$, then $|f(x)| \leq M$.

if necessary the proofs should involve the $\delta - \varepsilon$ definition of a limit.

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The problem had two previous portions to it that I already know how to do. However, you can reference them to do the posted portions of the problem. Here they are:

(a) Show that for each positive integer $n$ and each real number $x$, $f(nx)=nf(x)$.

(b) Suppose $f$ is such that there are $M>0$ and $a>0$ such that if $x\in [−a,a]$, then $|f(x)|\le M$. Choose $\varepsilon > 0$. There is a positive integer $N$ such that $M/N < \varepsilon$. Show that if $|x-y|