Proving the fundemental theorem of Calculus

I am currently trying to write a proof for the fundemental theorem of Calculus, mainly to practice writing proofs, and I have an attempt below. The problem is, I am unsure of how to interpret the result (and unsure how to adress certain assumptions). Also, I am not very well-versed in mathematics in general, so I might be using notation in a no-no fashion.

Let $f$ be a function which is continous on the interval $[a,b]$. For simplicity, we shall also assume that $f,f'\ge0$ over the given interval. Then, the set $P[a, b] =\{ (x,y):a\le x\le b,0\le y\le f(x) \}$ will be measureable with the following "area-measure":

$\displaystyle \mu(P[a,b])=\lim_{n \to \infty} \sum_{k=0}^{n}f(a+\frac{k(b-a)}{n})\cdot\frac{b-a}{n}$

Fix a point $a\lt c\lt$b. Then, the area between the point $x=c$ and some point $x=c+h$, for some fitting $h>0$, can be expressed by:

$\displaystyle \mu(P[a,c+h])-\mu(P[a,c])$

Because of the function's monotonicity, and the fact that it's non-negative, we can create the following inequalities:

$\displaystyle hf(c)\le\displaystyle \mu(P[a,c+h])-\mu(P[a,c])\le hf(c+h)$

Dividing by $h$ and applying a limit where $h\to 0$, one obtains:

$\displaystyle\lim_{h \to 0} f(c)\le \lim_{h \to 0} \frac{\mu(P[a,c+h])-\mu([P(a,c)])}{h}\le\lim_{h \to 0} f(c+h)$

Since both the LHS and the RHS simplfiy to $f(c)$, we can infer by the squeeze theorem that:

$\displaystyle \frac{\partial }{\partial c}\mu(P[a,c])=f(c)$

This differential equation has a class of solutions, namely:

$\mu(P[a,c]) = F(c)+C$, where C is some real constant.

I am unsure how to interpret this. The result I was expecting was something along the lines of $F(c)+C-(F(a)+C)=F(c)-F(a)$, but I am not sure how to arrive here.

Also, I made some assumptions in the beginning, such as $f,f'\ge 0$, mainly to get nice inequalities. I suppose if $f'<0$, one could simply flip the inequality symbols, but that seems clunky. Is there some more elegant way to do it?

Lastly, I am unsure whether I should be using an area measure and a point set here, or if it's better to just create a map $A:a,b\mapsto \lim_{n \to \infty} \sum_{k=0}^{n}f(a+\frac{k(b-a)}{n})\cdot\frac{b-a}{n}$. Not sure if it really matters, but I have never seen someone differentiate a measure like that. Frankly, I am not even sure if it's possible to do.