Consider the set $C([0, 1]) = \{f : [0, 1] \to \mathbb{R} : f$ is continuous }, with the supremum metric $d_{sup}$ $$d_{sup}(f, g) = sup{|f(x) - g(x)| : x \in [0, 1]}$$
Let $\phi : C([0, 1]) \to \mathbb{R}$ be the function given by the formula: $\phi(f) = f(0) + f(1)$
Prove that $\phi$ is continuous.
I don't know how to apply that metric to function $\phi$. Taking to $\phi$ different functions $f$ (that are continous on $x ∈ [0, 1]$) I will get different values at those ends of interval: f(0), f(1). But how /where do I use the supremum metric (that expresses an abstract category of distance between 2 random functions, but in my $\phi$ I have only one function and sum, not difference)? I am confused, how does $\phi$ look like whith that metric?
This is just like showing that any other function is continuous.
Fix $f\in C([0,1],\mathbb{R})$. Let $\varepsilon>0$ be given.
We need to find $\delta>0$ such that for $g\in C([0,1],\mathbb{R})$ with $d_{\sup}(f,g)<\delta$, we have $$|\phi(f)-\phi(g)|=|f(1)+f(0)-(g(1)+g(0))|\leq |f(1)-g(1)|+|f(0)-g(0)|<\varepsilon$$
Hopefully this puts you on the right track. I will leave it to you to choose the correct $\delta$.