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Calculus for mathematicians (1997) [pdf]

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Re: Calculus for mathematicians (1997) [pdf]

#5

Post by eccstartup »

bumbledraven wrote: post #3
eccstartup wrote: post #2

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

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

Are they equivalent? Or both systems work smoothly?

Re: Calculus for mathematicians (1997) [pdf]

#6

Post by dirkt »

eccstartup wrote: post #5
bumbledraven wrote: post #3

Earlier quoted context omitted.

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

Are they equivalent? Or both systems work smoothly?

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.

Re: Calculus for mathematicians (1997) [pdf]

#7

Post by bumbledraven »

eccstartup wrote: post #5
bumbledraven wrote: post #3

Earlier quoted context omitted.

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

Are they equivalent? Or both systems work smoothly?

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

Re: Calculus for mathematicians (1997) [pdf]

#8

Post by cottonseed »

dirkt wrote: post #6
eccstartup wrote: post #5

Earlier quoted context omitted.

Are they equivalent? Or both systems work smoothly?

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.

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

Re: Calculus for mathematicians (1997) [pdf]

#9

Post by anon4 »

hudibras wrote: post #4

Definitely stashing this away in my time machine for when I travel back to the 17th century. Make both Leibniz and Newton cry...

If you're taking things like that back in time, make sure to translate them to French. Also, learn French. Not knowing French in the 17th century is like not knowing English today.
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