Why is it wrong to prove the first fundamental theorem of calculus by geometric argument? I read from the Apostol book calculus Vol 1 that the First fundamental theorem of calculus is as follows:
Let $f$ be a function that is integrable on $[a,x]$ for each $x$ in $[a,b]$. let $c$ be such that $a\le c\le b$ and define a new function $A$ as follows:
$$A(x) = \int^x_c f(t)$$
if $a\le x\le b$. Then the derivative $A'(x)$ exists at each point $x$ in the open interval $(a,b)$ where $f$ is continuous, and for such $x$ we have
$$A'(x)=f(x)$$
The book gives a geometric argument as follows:
$$\int^{x+h}_x f(t)dt = \int^{x+h}_c f(t)dt - \int^{x}_c f(t)dt = A(x+h) - A(x)$$
Assuming $f$ is continuous in $[x,x+h]$, then by mean-value theorem for integrals, then we have $A(x+h) -A(x) = hf(z)$ where $x\le z \le x+h$.
The book told us that the above proof is flawed. What's wrong with this argument? Yes it assumes the continuity of interval $[x,x+h]$, but that's what the assumption of the theorem is about, right?
 A: As Apostol points out, this argument depends on the continuity of $f$ in the interval $[x,x+h]$ because one of the hypotheses of the mean value theorem for integrals is continuity on the whole interval of integration. It is flawed in the sense that it is not sufficient if all we assume is continuity of $f$ at $x$. Instead, we must assume continuity in a neighborhood of $x$.
I disagree with the comment above by Rob Arthan. At this stage of the book, Apostol has already developed more than enough theory concerning integrals. Though Apostol calls the argument "geometric" and gives an illustration with areas, the proof itself doesn't rely in any way on the geometric concept of area.
Edit: I will quote more of what Apostol says in that section of the book.

Hence we have $(A(x+h) - A(x))/h = f(z)$, and, since $x \leq z \leq x + h$, we find that $f(z) \to f(x)$ as $h \to 0$ through positive values. A similar argument is valid if $h \to 0$ through negative values. Therefore $A'(x)$ exists and is equal to $f(x)$.


This argument assumes that the function $f$ is continuous in some neighborhood of the point $x$. However, the hypothesis of the theorem refers only to continuity of $f$ at a single point $x$. Therefore, we use a different method to prove the theorem under this weaker hypothesis.

