$\mathrm{sign}(f) = \mathrm{sign}(\int_{\mathbb{R}}f(x) dx f) $?

Question

Under what conditions do we have $\mathrm{sign}(f) = \mathrm{sign}(\int_{\mathbb{R}}f(x) dx f) $, if $f$ does not have constant sign?

Answer 1

The only condition for that to hold is that

$$\int_{\mathbb R} f(x) dx > 0.$$

Answer 2

Assuming you have an integrable function and assuming $$ \int_{\mathbb{R}} f(x)dx=c \in \mathbb{R} $$ And so your problems becomes for $c\neq 0$ $$ sign(f) =sign(cf) $$ which is true if $c > 0$.
If $c<0$, the statement can never be true.
If $c=0$ then $f=0$, look up the defintion of the sign function for that one. If $f \neq 0$ the statement is also never true for all $x \in \mathbb{R}$.