Use the basic definition of continuity at c.
Use and find the that goes with it.
Then let .
Let f:[a, b] -> R be continuous at c of [a,b] and suppose that f(c) > 0. Prove that there exist a positive number m and an interval [u,v] is a subset of [a,b] such that c of [u,v] and f(x) >(or equal) for all x of [u,v].
I don't know where to begin with this problem!!
ok, so we know that f is continuous so,
and .
So to prove my thing do I go something like with with
You told me to find and I found it to be
I know this may sound stupid but I have starred at that since last night and I still can't figure out how to get to
Well, that was not a full proof but rather an outline more or less.
There are several careful adjustments to make to get u & v .
In the definition of continuity at c let .
Now corresponding to that epsilon .
Remove the absolute value: .
That means that every x within a distance of of c has the property .
Let .