is compact, so has a max on that interval.
Note: This only works in the reals. I notice you have as your space, so I'm not sure this applies, unless you know that closed intervals are compact.
If and , then F is continuous on
Proof so far.
Let ,
I need to show that , there exists such that whenever where , then I will have
Now, I know have that
Since f is integrable, this integral exists, but how should I refine it so it is less than ?
Thank you.
Be careful, where \delta is the Dirac delta function (1 at x_0, 0 everywhere else). Now the Dirac delta function has Lebesgue measure 0 (i.e. integral 0). The sequence is dominated by f (which is integrable), so DMC implies that the integral converges to 0, because DMC tells you that the integral of your sequence converges to .