I am amazed at seeing that the limit is defined by continuity. It is interesting...

bumbledraven · on May 28, 2015

99. Expository notes

Common practice in calculus books is to define continuity using limits. I define limits using continuity; continuity is a simpler concept.

eccstartup · on May 28, 2015

Are they equivalent? Or both systems work smoothly?

dirkt · on May 28, 2015

Continuity in the topological sense implies continuity by limits. For topological spaces with a countable (local) basis, the converse is also true.

So, in general it's not equivalent. For the reals etc., it is.

cottonseed · on May 28, 2015

They are equivalent if you consider limits of nets rather than limits of sequences.

bumbledraven · on May 28, 2015

They are logically equivalent for the reals. Search the web for [limits "in terms of continuity"].