Proof: By Induction
(related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
By hypothesis Let $I$ be a real interval and $f:I\to\mathbb R$ is a continuous function.
"$\Rightarrow$"
- Assume, $F,G:I\to\mathbb R$ are antiderivatives of $f.$
- By definition, this means for their derivatives that $F^\prime(x)=G^\prime(x)=f(x)$ for all $x\in I.$
- Therefore, for every $x\in I,$ $(F^\prime(x)-G^\prime(x))=\frac d{dx}(F-G)(x)=0$
- It follows from the derivative of a constant that $F$ and $G$ differ at most by a constant.
"$\Leftarrow$"
- Assume, $F(x)-G(x)=c$ is constant for all $x\in I.$
- Then, $G^\prime(x)=\frac d{dx}(F(x)-c)=F^\prime(x)=f(x)$ for all $x\in I.$
- It follows that $G$ and $F$ are antiderivatives of $f.$
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