How to show that $(C^0((a,b)), d_\infty)$ is not a metric space

Question

Let $d_\infty:C^0([a,b]) \times C^0([a,b]) \to [0,\infty)$ be defined as

$$ d_\infty(f,g)=\sup\limits_{x \in [a,b]} \left\{ |f(x) - g(x)| \right\} $$

I have already shown that $(C^0([a,b]), d_\infty)$ is a metric space.

How can I show that $(C^0((a,b)), d_\infty)$ is not a metric space? I'm confused as I did not use the compactness of the intervall when proving that $(C^0([a,b]), d_\infty)$ is a metric space.

Answer 1

Hint: If $\displaystyle f : x \mapsto \frac{1}{x-a}$, what is $d_{\infty}(f,0)$?