I want to show that if the integral of f with upper limit b and lower limit a is equal to 0
then f(x) = 0 for all x in [a,b]?
Which is clearly not true,
the identity function on
Perhaps, you wish to show that if
And if,
Then,
But that is also not true.
Consider an analoge of the Dirac Delta Function.
That is, the function is discontinous at one point with a different value and elsewhere is equal to zero.
yea thats wat i wanted to show if f(x)>0 and if the integral of f= 0 with upper limit b and lower limit a then f(x)= 0 for all x in [a,b]...but i dont get what this has to do with the qiestion "Consider an analoge of the Dirac Delta Function.
That is, the function is discontinous at one point with a different value and elsewhere is equal to zero"
Consider the function,
on and for
Then, the integral still exists,
though it is discontinous on this interval.
Yet,
for all the points in
Perhaps, you want to show and is continous on the interval. (Look how many conditions you have to fullfil to demonstrate what you have said!)