If $f^{\prime}(x)=g^{\prime}(x)$ at each point $x$ in an open interval $(a, b)$, then there exists a constant $C$ such that $f(x)=g(x)+C$ for all $x \in(a, b) .$ That is, $f-g$ is a constant function on $(a, b)$ #### Proof At each point $x \in(a, b)$ the derivative of the difference function $h=f-g$ is $ h^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x)=0 $