Integral of two functions which differs to each other in a point

Question

I'm dealing with a base exercise of analysis: given $$f,g:[a,b]\to \mathbb{R}$$ such that $f(x)=g(x)$ for all $x \in [a,b]$ but not for a certain $x_{0}\in [a,b]$ then $$\underline{\int}_{a}^{b}f(x)dx=\underline{\int}_{a}^{b}g(x)dx$$ I know that it comes as a corollarium of more general theorems but I have to prove this directly by using Darboux's integration.

Answer 1

Let $\alpha=|f(x_0)-g(x_0)|$. Choose a partition $P$ such that $mesh \,P<\epsilon$ for some $\epsilon>0$. Then $|\mathcal{L}(f-g,P)|<\alpha\epsilon$. This holds for all refinements of $P$. Since $\epsilon>0$, is arbitrary, and $\alpha$ is fixed, the result follows.