Let f be a function which is continous on a closed interval [tex][a,b][/tex] with [tex]f(c) > 0[/tex] for some [tex]c\in[a,b][/tex]. Show that there is a closed interval [tex][r,s][/tex] with [tex]c\in[r,s]\subseteq[a,b][/tex] such that f(x) > 0 for all [tex]x\in[r,s][/tex].
Hint let epsilon = f(c)/2 and find [tex]\delta > 0[/tex] such that |f(x) - f(c)| < [tex]\epsilon[/tex] when |x - c| < [tex]\delta[/tex]
The Attempt at a Solution
If there is a f(c) with c within [a,b], that is greater than zero, then by continuity there must be an interval for which the function is greater than 0, [r,s], that seems fairly obvious.
By completeness [r,s] must have a least upper bound, and a least lower bound this would be x when f(x) = 0 (not sure if that helps or not).
I am not really sure what the hint is telling me at this stage, or why one would pick f(c)/2. I am guessing one could show that there is a range for which f(x) > f(c)/2 or some other constant. It seems as though one could repeat this process and get closer and closer to the edge of the interval.
I am not sure if i am on the right track, but I'd appreciate a bit of input on putting things together, assuming I am even on the right track.