I can do (a) and (b), but I cannot do (c). I have attempted to use $$f_n(x)=\frac{f(x+\frac{1}{n})-f(x)}{\frac{1}{n}},$$ but I do not know how to proceed.
(a) If $f(x) \in L^2(X)$ and $\mu(X)<\infty$, show that $f(x) \in L^1(X)$.
(b) If $f(x)$, $x f(x) \in L^2(\mathbb{R})$, show that $f(x) \in L^1(\mathbb{R})$.
(c) If $f(x)$, $g(x) \in L^2(\mathbb{R})$ and $$\lim _{h \rightarrow 0} \int_{\mathbb{R}}\left|f_h(x)-g(x)\right|^2 d x=0,$$ where $$f_h(x):=\frac{f(x+h)-f(x)}{h} \quad \text { for any } h \in \mathbb{R} \setminus \{0\},$$ show that $$f(x)=\int_{[0, x]} g(t) d t+C $$for some constant $C$.