The continuity of [tex]f'[/tex] and its limiting behavior guarantees that [tex]f'[/tex] is Riemann integrable, so you can write
[tex]\displaystyle\int_0^\infty f'(x)\,\mathrm dx=uv\bigg|_{x=0}^{x\to\infty}-\int_0^\infty v\,\mathrm du[/tex]
where [tex]u=1\implies\mathrm du=0\,\mathrm dx[/tex] and [tex]\mathrm dv=f'(x)\,\mathrm dx\implies v=f(x)[/tex], so that
[tex]\displaystyle\int_0^\infty f'(x)\,\mathrm dx=f(x)\bigg|_{x=0}^{x\to\infty}=\lim_{x\to\infty}f(x)-f(0)=-f(0)[/tex]
