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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