Thread: Question on proving the property of an integral

Hi,

I have a question which I'm having trouble with.

I am told that I should first use the epsilon delta definition of continuity to prove that there must be a closed interval [c,d] with c < x0 < d such that for some x in [c, d], |h(x)| > |h(x0)|/2 by choosing epsilon to be |h(x0)|/2. I am having trouble with this.

Any help would be appreciated

Just instantiate epsilon to |h(x0)| / 2 in the definition of continuity of h at x0. What do you get?

Errr....well h is continuous at x0 when if |x - x0| < delta, then ||h(x)| - |h(x0)|| < |h(x0)|/2. Sorry but you might have to spell it out for me....

Which, in turn, means that -h(x0)/2< h(x)- h(x0)< h(x0)/2. (You don't need absolute value on h(x0) because they are by hypothesis positive) and the absolute values on h(x) and h(x0) on the left are wrong.)

Now add h(x0) to each part. h(x0)- h(x0)/2= h(x0)/2< h(x)< h(x0)/2+ h(x0)= 3h(x0)/2. And, again, by hypothesis h(x0) is positive.

yep, made sense of it. Thanks for your help