> [!NOTE] Theorem (The Fundamental Theorem of Calculus II) > Let $f$ be a [[Real Function|real function]] that is [[Riemann Integration|Darboux integrable]] on a [[Closed Real Interval|closed real interval]] $[a,b].$ Let $F:[a,b]\to \mathbb{R}$ be a real function that is [[Continuous Real Function|continuous]] on $[a,b]$ and [[Fréchet Differentiation|differentiable]] on $(a,b)$ such that its [[Derivative of Real Function|derivative]] satisfies $F'=f.$ Then we have $\int_{a}^{b} f(t) \, dt = F(b) - F(a). $ **Proof**: STS that for all [[Finite Partition of Closed Real Interval|finite partitions]] of $[a,b],$ $P,$ we have that the [[Riemann Integration|lower]] and [[Riemann Integration|upper Darboux sums]] of $f$ wrt $P$ satisfy $L(f,P)\leq F(b)-F(a)\leq U(f,P)$because we then get the [[Riemann Integration|upper]] and [[Riemann Integration|lower Darboux integrals]] satisfy $\underline{\int}f \leq F(b)-F(a)\leq \overline{\int} f$and if $f$ is integrable the upper and lower sums are equal. Let $P=\{ x_{0}, x_{1},\dots,x_{n} \}$ be a finite partition of $[a,b].$ For each $i=1,2,\dots,n,$ $F$ is continuous on $[x_{i-1},x_{i}]$ and differentiable on $(x_{i-1},x_{i})$ so by [[Mean value theorem|MVT]] there is a point $c\in(x_{i-1},x_{i})$ with $F(x_{i})-F(x_{i-1})=f(c_{i})(x_{i}-x_{i-1}).$Hence for each $i,$ $m_{i}(x_{i}-x_{i-1})\leq F(x_{i})-F(x_{i-1})\leq M_{i}(x_{i}-x_{i-1})$where $m_{i}$ and $M_{i}$ are the [[Supremum of Set of Real Numbers|supremum]] and [[Infimum of Set of Real Numbers|infimum]] of the [[Image of a set under a function|image]] $f([a,b]).$ Summing over $i$ gives $L(f,P)\leq F(b) - F(a) \leq U(f,P).$