I will give you some hints for part b.
Because f is continuous, there is a subinterval of [a,b] on which
To see how this works note that:
If f(x)>0 for all xElement-ofsymbol [a,b], then
b
Integral sign f>0
a
a) Give an example where f(x)>0 for all xElement-ofsymbol [a,b], and f(x)>0 for some xElement-ofsymbol [a,b], and
b
Integral sign f=0
a
b) Suppose that f(x)>0 for all xElement-of symbol[a,b] and f is continous on at x0 in [a,b] and f(x0)>0. Proove that
b
Integral sign f>0
a
Have you ever heard of the "Limit-Limitation Theorem".If , .then: .
a) Give an example where , and ,
. . and
b) Suppose that and is continous at , and .
Prove that: .
It states that if and exists then the limit value is non-negative. (Note can also be limits at infinity).
Now, the integral is the limit of the Riemann sum. The rest is trivial.
For you other question,
Consider,
This function is not always zero.
And because it is continous almost everywhere,