What does $sup\{|f_1(x)-f(x)|, x \in D\}$ exactly mean?

Question

I think it means the maximal distance between $f1$ and $f$ when you try all $x's$, I mean you test the distance between $f1$ and $f$, for all $x's$ and then you pick the max aka the supremum. Is this right?

Answer 1

Yep, you're right. The supremum (as you seem to have got down) is like the max, except it doesn't need to be in the set.

For example,

$$\sup{(\{\frac{x}{x+1} \mid x \in \mathbb{R}\})} = 1$$

but $1$ is not in that set.