The fundamental theorem identifies differentiation and integration as inverse operations, provided the correct regularity is used. In modern analysis, absolute continuity is the natural hypothesis.
Integral functions are absolutely continuous
Integrating an \(L^1\) function produces an absolutely continuous primitive.
Theorem. If \(g\in L^1([a,b])\) and \(F(x)=\int_a^x g(t)\,\mathrm{d}t\), then \(F\) is absolutely continuous and \(F'=g\) almost everywhere.
Proof. Given \(\varepsilon>0\), absolute continuity of the Lebesgue integral supplies \(\delta>0\) such that \(m(E)<\delta\) implies \(\int_E|g|<\varepsilon\). For disjoint intervals with total length below \(\delta\), \[ \sum_k|F(b_k)-F(a_k)| \le\int_{\cup_k(a_k,b_k)}|g|<\varepsilon. \] Thus \(F\) is absolutely continuous. At a Lebesgue point \(x\) of \(g\), \[ \frac{F(x+h)-F(x)}h =\frac1h\int_x^{x+h}g(t)\,\mathrm{d}t\longrightarrow g(x) \] as \(h\to0\), with the oriented interpretation for \(h<0\). Almost every point is a Lebesgue point.
Recovering the function from its derivative
Absolute continuity is the exact condition needed for the derivative to reconstruct the function.
Theorem. If \(F\) is absolutely continuous on \([a,b]\), then \(F'\in L^1\) and \[ F(x)-F(a)=\int_a^x F'(t)\,\mathrm{d}t. \]
Proof. Absolute continuity gives a finite signed measure \(\nu((s,t])=F(t)-F(s)\) satisfying \(\nu\ll m\). By Radon-Nikodym, \[ \nu(E)=\int_Eh\,\mathrm{d}t \] for some \(h\in L^1\). In particular, \[ F(x)-F(a)=\nu((a,x])=\int_a^xh(t)\,\mathrm{d}t. \] The first part of the theorem gives \(F'=h\) almost everywhere, proving both integrability of \(F'\) and the reconstruction formula.
Corollary. An absolutely continuous \(F\) is constant if and only if \(F'=0\) almost everywhere. If \(F'\ge0\) almost everywhere, then \(F\) is increasing.
Proof. For \(x<y\), \[ F(y)-F(x)=\int_x^yF'(t)\,\mathrm{d}t. \] The two conclusions follow from vanishing or nonnegativity of the integrand.
Classical substitution
Change of variables transforms integrals by the local stretching factor.
Theorem. If \(\phi:[\alpha,\beta]\to[a,b]\) is \(C^1\) and \(f\) is continuous, then \[ \int_{\phi(\alpha)}^{\phi(\beta)} f(u)\,\mathrm{d}u=\int_\alpha^\beta f(\phi(t))\phi'(t)\,\mathrm{d}t \] when \(\phi\) is monotone, with orientation included.
Proof. Let \(F(x)=\int_{\phi(\alpha)}^x f(u)\,\mathrm{d}u\). Then \(F'=f\), and by the chain rule \((F\circ\phi)'=f(\phi(t))\phi'(t)\). Integrating from \(\alpha\) to \(\beta\) gives the identity.
Monotone absolutely continuous substitutions
For a monotone map, the derivative accounts for local stretching even when the inverse is not differentiable.
Theorem. If \(\phi:[a,b]\to\mathbb R\) is increasing and absolutely continuous, then for every nonnegative measurable \(f\), \[ \int_a^b f(\phi(x))\phi'(x)\,\mathrm{d}x =\int_{\phi(a)}^{\phi(b)}f(y)\,\mathrm{d}y. \] The identity also holds for signed \(f\) whenever one side is absolutely integrable.
Proof. For \(f=\mathbf1_{(u,v]}\), the preimage under increasing \(\phi\) is an interval, up to endpoints and flat pieces. Applying the fundamental theorem to \(\phi\) on that interval gives \[ \int_{\{\phi\in(u,v]\}}\phi'(x)\,\mathrm{d}x =m\bigl((u,v]\cap(\phi(a),\phi(b)]\bigr). \] Linearity proves the formula for nonnegative simple functions. Choose simple \(f_n\uparrow f\) and apply monotone convergence to both sides. Finally decompose an integrable signed function into positive and negative parts.
Multiplicity in the nonmonotone area formula
Theorem (one-dimensional area formula). If \(\phi\) is Lipschitz on \([a,b]\) and \(g\ge0\) is measurable, then \[ \int_a^b g(x)|\phi'(x)|\,\mathrm{d}x =\int_{\mathbb R}\sum_{x\in\phi^{-1}(y)}g(x)\,\mathrm{d}y. \] In particular, with \(g(x)=f(\phi(x))\), \[ \int_a^b f(\phi(x))|\phi'(x)|\,\mathrm{d}x =\int_{\mathbb R}f(y)N(\phi,y)\,\mathrm{d}y, \] where \(N(\phi,y)\) counts preimages.
Proof outline. Decompose the interval, up to a null set, into countably many measurable pieces on which \(\phi\) is one-to-one and monotone. Apply the monotone substitution formula on each piece and sum by Tonelli's theorem. Each value \(y\) appears once for every inverse branch, producing the multiplicity \(N(\phi,y)\).
Where formulas fail
Without absolute continuity, derivatives may miss singular variation.
Remark. The Cantor function has derivative zero almost everywhere but is not constant. Thus the formula \(F(b)-F(a)=\int_a^bF'\) requires absolute continuity, not merely differentiability almost everywhere.