prove function f has boundry in [a,b],if f is continuous in [a,b], using finite covering, and compact theorem

prove:
by "contiunuous function has local boundry",\(\forall x\in [a,b]\)
exists a \(\delta\) and M, for all \(x\in\U(x;\delta)\),x\(\leqslant<M\)
the set H={U;x}