S.O.S. Mathematics CyberBoard

Suppose that f' is continuous in a neighborhood of c and f'(c)>0. Then f is strictly increasing in some neighborhood of c.

I'm having trouble determining this from the information given. I feel as if I can give an obvious counterexample, but the "SOME neighborhood of c" part makes me wary. Any help appreciated.

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let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

Some neighborhood implies that the neighborhood be small enough that f'(x) > 0 in the neighborhood. There is such a neighborhood, since f'(c) > 0 and f'(x) is continuous.

Might this be relevant?

http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

Not particularly, as is noted in the article, if is continuous, then it just follows from the MVT, which is what you really want. Since is continuous, we know that the inverse image of an open set around is an open set surrounding , but then if we take , then this is clearly an open neighborhood of which contains only positive numbers since , so its inverse image is the desired neighborhood of .

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This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Ok, that makes sense, the problem seemed to have an IVT ring to it, but I didn't know if that was strong enough...unlikely since was an explicit hypothesis. I will see just for kicks if I can work out a proof by contradiction.

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

Any direct proof can be made into a proof by contradiction, just assume you're wrong, and go through the usual proof, you arrive at the truth, which is the contradiction to the falsehood. True proofs by contradiction involve existence theorems which don't produce an explicit result. However, this should be possible for you, at least in theory, since the IVT and MVT are not constructive theorems.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Thank you both for your replies.

Shadow-
That was a fantastic insight and very helpful. Thanks!

_________________
let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2