jimlucky的部落格

In Calculus,We argue continuity ==> limit exist
But it`s not right absolutely.
To consider isolated point(P) in domain.
then for any eplision..
Beacuse have no element in domain to approach to P.
so it`s clearly get function is continuous at P
But limit doesn`t exist.
so continuity ==> limit exist only adapt when p is limit point.
Give an another idea
consider definition of discontinuity.
~(for any eplison >0 exist r>0 such that d(x,a) is equivalent to Exist eplision >0 for any r>0 d(x,a)=eplision)
since d(f(x),f(a))>=eplision ,we know f(x)=/=f(a)
by defn of function ,x=/=a.
then for any r>0 d(x,a) so when we consider discontinuity.
we only have to consider limit point of domain.