I am unsure on a specific step in the proof of the corollary of FTC from Wikipedia. Suppose $F$ is an antiderivative of $f$ on the interval $[a,b]$. Wiki used the fact that $G$ define as $$G(x) = \int_a^x f(x) \;dx$$ is an antiderivative of $f$ to show that $$F(a)+c = G(a) \tag{1}$$ for some constant $c$. However, in wiki's proof of Fundamental Theorem of Calculus, we only concluded that $G$ as defined above is an antiderivative of $f$ on the interval $(a,b)$. Since $G$ might not be an antiderivative of $f$ on the point $a$, how can we use the property of antiderivative differ from each other by a constant to conclude equation (1)? Is there any other proof of FTC that shows $G$ is an antiderivative of $f$ on point $a$ and $b$?
Thank you so much in advance for answering this! It has bother me for a while now and I can't figure it out.
Best Answer
Long story short, we can use the continuity of $G(x)$ on $[a,b]$ to continuously extend the domain of $F(x)$ to include $a$!
By the first part of the fundamental theorem, we know that $G(x)$ is the antiderivative of $f(x)$ on $(a,b)$ and that $G(x)$ is continuous on $[a,b]$. Thus, while wikipedia doesn't specify the domain of the antiderivative $F(x)$, we should assume that it is at least defined as an antiderivative of $f(x)$ on the interval $(a,b)$ (we know that this much is possible since $G(x)$ exists). The argument that wikipedia presents then works for the interval $(a,b)$: $F'(x)-G'(x)=f(x)-f(x)=0$ on $(a,b)$ so $F(x)-G(x)$ must be some constant $c$ on $(a,b)$. We can then use the fact that $G(x)=F(x)+c$ on $(a,b)$ and the fact that $G(x)$ is continuous on $[a,b]$ to conclude that $\lim_{x\to a^+}F(x)$ must exist (if this limit did not exist, the relationship $G(x)=F(x)+c$ on $(a,b)$ would make $G(x)$ discontinuous at $a$). We then define $F(a)=\lim_{x\to a^+}F(x)$ and note that this gives us $G(a)=F(a)+c$.
This is a great detail to notice and take issue with! The way the article as written is not clear on this point.