Primitives which Differ by Constant

Theorem
Let $F$ be a primitive for a real function $f$ on the closed interval $\closedint a b$.

Let $G$ be a real function defined on $\closedint a b$.

Then $G$ is a primitive for $f$ on $\closedint a b$ :
 * $\exists c \in \R: \forall x \in \closedint a b: \map G x = \map F x + c$

That is, $F$ and $G$ differ by a constant on the whole interval.

Necessary Condition
Suppose $G$ is a primitive for $f$.

Then $F - G$ is continuous on $\closedint a b$, differentiable on $\openint a b$, and for any $x \in \openint a b$, we have:

From Zero Derivative implies Constant Function it follows that $F - G$ is constant on $\closedint a b$, hence the result.

Sufficient Condition
Now suppose $\map G x = \map F x + c$.

We compute:

Hence $G$ is also a primitive for $f$.