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Let $f$ be continuous on $\mathbb{R}$. Due to FTC I, we know that a function of the form∗ $F(x) = \int_a^xf(t)\operatorname dt$ is always an antiderivative of $f(x)$. In this question you will investigate whether all antiderivatives of $f(x)$ can be expressed in this form∗. For simplicity, let us further assume $f$ is non-negative $(i.e. ∀x ∈ \mathbb{R}, f(x) ≥ 0)$. (a) Suppose$\lim\limits _{A\rightarrow\infty}\int_0^Af(t)\operatorname dt$ or $\lim\limits _{A\rightarrow-\infty}\int_A^0f(t)\operatorname dt$ is finite, show there is an antiderivative $G(x)$ of $f(x)$ which does not equal $\int_a^xf(t)\operatorname dt$ for any a $\in \mathbb{R}$ (b)Suppose $\lim\limits _{A\rightarrow\infty}\int_0^Af(t)\operatorname dt=\infty$ and $\lim\limits _{A\rightarrow-\infty}\int_A^0f(t)\operatorname dt=\infty$, show for any antiderivative $G(x)$ of $f(x), ∃a ∈ \mathbb{R} \text{ s.t.} G(x) = \int_a^xf(t)\operatorname dt$ Hint: Think about whether antiderivatives of f(x) need to have zeroes.

What I have tried so far:

Look thorugh (a) and (b), it's saying if $f$ is continuous on $\mathbb{R}$ we have:

$(\lim\limits _{A\rightarrow\infty}\int_0^Af(t)\operatorname dt=\pm\infty \wedge \lim\limits _{A\rightarrow-\infty}\int_A^0f(t)\operatorname dt=\pm\infty )\leftrightarrow \forall G(x), ∃a ∈ R \text{ s.t.} G(x) = \int_a^xf(t)\operatorname dt$

(This is a stronger version of the question, since negation of finite also include $-\infty$, I'm not sure if this is still true, but this should implies what the question is asking to prove)

By assumption, $f$ is non-negative, then we don't need to consider the $-\infty$ cases, just show the following would be sufficient:

$(\lim\limits _{A\rightarrow\infty}\int_0^Af(t)\operatorname dt=\infty \wedge \lim\limits _{A\rightarrow-\infty}\int_A^0f(t)\operatorname dt=\infty )\leftrightarrow \forall G(x), ∃a ∈ R \text{ s.t.} G(x) = \int_a^xf(t)\operatorname dt$

I don't have the Intuition of why this is true, at least it's not very trivial to me..

So, first I tried to break it into definitions:

1.$\lim\limits _{A\rightarrow\infty}\int_0^Af(t)\operatorname dt=\pm\infty$

$\Leftrightarrow \forall N\in \mathbb{R},\exists M\in \mathbb{R} s.t. A>M\rightarrow(\int_0^Af(t)\operatorname dt>N\vee \int_0^Af(t)\operatorname dt<N)$

2.$\lim\limits _{A\rightarrow-\infty}\int_A^0f(t)\operatorname dt=\pm\infty$

$\Leftrightarrow \forall N\in \mathbb{R},\exists M\in \mathbb{R} s.t. A<M\rightarrow(\int_0^Af(t)\operatorname dt>N\vee \int_0^Af(t)\operatorname dt<N)$

3.$\forall G(x), ∃a ∈ R \text{ s.t.}G(x) = \int_a^xf(t)\operatorname dt$ (not sure about this one)

$\Leftrightarrow\forall G(x), ∃a ∈ R \text{ s.t.}\forall n \in \mathbb{R}, \forall \varepsilon>0, \exists \delta>0\text{ s.t. } \exists P\in \mathbb{P}$ s.t.

$( \text{$P$ is a partition of [a,n]} \wedge l(P)<\delta)\rightarrow|S(f(t),P)-G(n)|<\varepsilon$

But those doesn't looks like very useful...where should I start?

Any help or hint or suggestion would be appreciated.