Why is lim \int (a,x) f(x) dx not equal to 0?

[tomboi03]

Prove: If f is integrable on [a,b] then
lim \int(a,x) f=0

I put If F is integrable for every E>0, partition,P, U(f,P) - L(f,P) <E
This also means that f is integrable on [a,a+]

If \int (x,x) f(x) dx =0
F(x)-F(x)=0
Since f is integrable, if can be differentiable (he said that this was wrong)
lim \int (a,x) f(x) dx
x\rightarrowa+

lim \int (a,x) f(x) dx
x\rightarrowa

lim \int (a,x) f(x) dx
x\rightarrowa-

are all equal then
lim \int (a,x) f(x) dx
x\rightarrowa+


what am i doing wrong?

 

[mutton]

"[a, a+]" is not an interval.

The limits of integration should not depend on x.

Integrability does not imply differentiability. For example, |x| is integral on [-1, 1] but not differentiable at 0.

It's not clear how the parts of your proof are connected.