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.
Might this be relevant?http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)
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