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Proving there is a constant $C$ such that $f=g+C$

Consider the following lemma:

Lemma: Let $f:[a,b]tomathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, If $f'(x)=0$ for all $xin [a,b]$, then $f$ is constant on $[a,b]$.

I have to prove the following corollary:

Corollary: If $f$ and $g$ are continuous functions on $[a,b]$, differentiable on $(a,b)$, and $f'(x)=g'(x)$ for all $xin(a,b)$, then there exists a constant $C$ such that $f=g+C$.

My attempt: Consider the function $h(x)=f(x)-g(x)$. We have that $h'(x)=f'(x)-g'(x)=0$ for all $xin(a,b)$. By the lemma we have that $h$ is constant on $[a,b]$. In other words, $h(x)=C$, where $Cinmathbb{R}$ is a constant. Then, $C=f(x)-g(x)$ in $[a,b]$. We conclude that $C+g(x)=f(x)$.

Is this proof correct?

Mathematics Asked by user926356 on December 30, 2020

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