Fundamental lemma in calculation of variation states that $\int_a^b f(x)g(x) dx =0$ for $f$ continuous in $(a, b) $ and and $g(x)$ which has compact support and is continuous in $(a,b)$ , then $f(x)=0$ $\forall x\in (a,b)$ . I found out the proof of this theorem is easy . But i have some doubts related to it if i want to extend this concept in heigher dimensions and to different class of functions .
a) Instead of interval $(a,b) $ if i take a set $\Omega \subset R^n$ , then what do i have to think about here, to prove the above theorem ?
b) Is it necessary that $g(x)$ should be infinitely differentiable and compactly supported ?
c) If $\int_a^b f(x) g'(x) =0$ with $g(x) , f(x) \in C^1[a,b]$ and $g(a)=g(b)=0$ . Then $f(x)$ is constant but does it still remain the same if i take $f(x)\in C^0[a,b]$ and $g(x) \in C_c^1[a,b]$ ?
A bit vague question : If i want to extend this idea to large class of functions what are the important things that i need to go about ? Thank you for your help .
What i am thinking about $c$ is to use mollification but the problem is it will be valid only almost everywhere, which i am not interested . is it possible after all ?