Question

If anyone could help with proving that f: D->R has a limit a at x in D if and only if f has a limit from above and a limit from below at x and both coincide, that would be appreciated.

Answer 1

The function is mapping the Domain onto the Range. f:D->R.

Proofing tho..... not my strong point

Answer 2

The most I can add to this question:
Lim of the function as x tends to a from the left is equal to the lim of the function as x tends to a from the right...this means that you work both for the function and derive that both their limits (ie from the left and from the right) exist and are equal

Answer 3

maybe throw in a couple of deltas and epsilons to be on the safe side :)

Answer 4

Consulting definitions to terms in the problem is usually the first approach. Maybe the definitions for conincide, limits above and below, etc.

You can also start out from the back and work your way toward the original question. I.e. a good question to ask would be how can I show that both limits above and below coincide.

Also since this is an implication in both directions (i.e. if and only if) you'll have two things:

a) If a function with domain D mapped onto real numbers (I'm assuming that's what R means) has a limit at x in D then there is a limit above and below at x and they coincide.
(foreward direction)

b) If there is a limits above and below that coincide at x in the function f, then there is a limit at x in D.
(backward direction)

You might want to think about this graphically as well.

I know this doesn't give the answer, but it may help you think about the problem.

Answer 5

thanks very much for your help guys, ill use your advice to do my best and question my lecturer about it tommorow.