Is the supremum a function of a function just the maximum

Question

Is

$$ \sup_{x\in S} f(x) \equiv \max_{x\in S} f(x) $$

And is the correct definition of a maximum of a function the following: $$ \max_{x\in S} f(x):= y\quad \text{such that}\ \exists x\in S,y=f(x)\wedge\forall x\in S, f(x)\le y $$