Let $f:\mathbb{R}\to \mathbb{R}$ a continuous function. Does exist a function $F:\mathbb{R}\to \mathbb{R}$ such that $F'(x) = f(x)$ $\forall x \in \mathbb{R}$? In compact intervals is clear, because of the Fundamental Theorem of Calculus. Can this be extended to the whole real line? In other words, let $a\in \mathbb{R}$. Consider $$ F(x) = \int_a^x f(t) dt $$ If $a<x$, then $F'(x) = f(x)$, because of FTC. What if $a>x$?
Thoughts: let $b<x$. Then $\int_a^x f(t) dt - \int_b^x f(t) dt \in \mathbb{R}$. The second integral $G(x) = \int_b^x f(t) dt$ verifies $G'(x) = f(x)$ $\forall x>b$, so is a primitive of $f$. Since $F(x)$ differs by a constant from a primitive in $(b,\infty)$, then is also a primitive in $(b,\infty)$.
Is my reasoning ok?