$\int_{\mathbb{R}}f(x)+g(x) \;dx<+\infty$ iff $ \int_{\mathbb{R}}g(x)\;dx<+\infty$ and $\int_{\mathbb{R}}f(x)\;dx<+\infty .$

Question

Let be $f,g\in C^0(\mathbb{R})$ such that $g,f\geq0$.

$$ \int_{\mathbb{R}}f(x)+g(x) \;dx<+\infty\quad {\rm iff} \quad\int_{\mathbb{R}}g(x)\;dx<+\infty\quad {\rm and}\quad\int_{\mathbb{R}}f(x)\;dx<+\infty .$$

When the functions are positive, this statement is true. Right?

Whithout $f,g\geq0$, is the implication $\Rightarrow$ true? I think that the answer is no, because I can find the indetermination $-\infty+\infty$.

Thanks for the clarification!