Full question:
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Prove that if $f :D\rightarrow \mathbb{R}$ is continuous, where $D$ is a compact set in $\mathbb{R}$, and
if $f$ is never zero on $D$ (so $f(x)\neq 0$ for all $x\in D$), then 

(1) there is a constant $M > 0$ such that $|f(x)|\geq M$ for all $x\in D$, and 

(2) $1/f(x)$ is continuous on $D$.

My attempt:
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I am struggling really hard on how to start the problem. The logic makes sense: if $|f(x)|$ and $M$ are both greater than zero, then we can choose M to be incredibly small such that it is less than or equal to $|f(x)|$. However, I cannot think of how to approach the formal proof.