- #1
stunner5000pt
- 1,461
- 2
i need some help with this proof. Please do not give me the solution i want to figure this on my own. ALl i need is your hints and steering.
Also note that i am on a first year level intro to real analysis course. I do however know the upper and lower reimann sums which is probably how this problem is to be solved.
Suppose f is continuous and non negative on [a,b]. Suppose f(c) > 0 for some c in (a,b). Prove that [tex] \int_{a}^{b} f > 0 . [/tex]
What worries me here is that f(c) does not include f(a) and f(b) exclusively. however how would i go about proving that? Would i have to use limits to show that for some epsilon [tex] \lim_{\epsilon \rightarrow 0} f(x - \epsilon) > 0 [/tex] and the same would apply for the f(b) paart??
I know that if i picked a partition P = {a,a+E,b-E,b} i would encounter this problem because of the explicit value (limit rather) of function at a nad b not being greater than zero.
P.S. i can't find a website that shows how to prove limits formally (although i have learned it in the past i cannot find it in my notes).
Also note that i am on a first year level intro to real analysis course. I do however know the upper and lower reimann sums which is probably how this problem is to be solved.
Suppose f is continuous and non negative on [a,b]. Suppose f(c) > 0 for some c in (a,b). Prove that [tex] \int_{a}^{b} f > 0 . [/tex]
What worries me here is that f(c) does not include f(a) and f(b) exclusively. however how would i go about proving that? Would i have to use limits to show that for some epsilon [tex] \lim_{\epsilon \rightarrow 0} f(x - \epsilon) > 0 [/tex] and the same would apply for the f(b) paart??
I know that if i picked a partition P = {a,a+E,b-E,b} i would encounter this problem because of the explicit value (limit rather) of function at a nad b not being greater than zero.
P.S. i can't find a website that shows how to prove limits formally (although i have learned it in the past i cannot find it in my notes).