Proof of Morera's theorem by a local primitive

Builds on Contour integral Holomorphic function Complex differentiability Integration by substitution Estimation lemma Holomorphic functions are infinitely differentiable

Proof

Let $U \subseteq \C$ be open and let $f : U \to \C$ be continuous. Assume $\oint_{\partial T} f(z) \, dz = 0$ for each closed triangle $T \subseteq U$ .
For $a, b \in \C$ , let $[a, b]$ be the segment path $t \mapsto a + t(b - a)$ , $t \in [0, 1]$ . All integrals below are contour integrals.
Reversing a segment changes the sign: $\int_{[b, a]} f = -\int_{[a, b]} f$ . This follows from the substitution $t \mapsto 1 - t$ .
Fix $c \in U$ and $R > 0$ with $B = \set{z : \abs{z - c} < R} \subseteq U$ . It suffices to show that $f$ is holomorphic on each such disc $B$ .
For $z \in B$ , define $F(z) = \int_{[c, z]} f(w) \, dw$ . The segment $[c, z]$ lies in $B$ because $B$ is convex.
Let $z \in B$ and let $h \ne 0$ with $z + h \in B$ . The triangle $T$ with vertices $c$ , $z$ , $z + h$ lies in $B$ , by convexity.
Claim: $\int_{[c, z]} f + \int_{[z, z + h]} f + \int_{[z + h, c]} f = 0$ . If the vertices are not collinear, the left side is $\pm \oint_{\partial T} f$ , so it is $0$ by step 1.
If the vertices are collinear, they lie on a line $s \mapsto c + s u$ with $\abs{u} = 1$ . By substitution, each segment integral becomes $\int f(c + s u) \, u \, ds$ over an interval of $s$ . The three intervals add to zero by additivity of the integral, which proves the claim.
By steps 3 and 7, $F(z + h) - F(z) = \int_{[z, z + h]} f(w) \, dw$ .
Also $\int_{[z, z + h]} f(z) \, dw = \int_0^1 f(z) \, h \, dt = f(z) \, h$ . So $\frac{F(z + h) - F(z)}{h} - f(z) = \frac{1}{h} \int_{[z, z + h]} \bigl( f(w) - f(z) \bigr) \, dw.$
Let $\varepsilon > 0$ . Since $f$ is continuous at $z$ , there is $\delta > 0$ with $\abs{f(w) - f(z)} < \varepsilon$ for $\abs{w - z} < \delta$ .
For $0 < \abs{h} < \delta$ , each $w$ on $[z, z + h]$ satisfies $\abs{w - z} < \delta$ . The segment has length $\abs{h}$ .
By the estimation lemma and step 12, the right side of step 10 has modulus at most $\frac{1}{\abs{h}} \cdot \varepsilon \abs{h} = \varepsilon$ .
So $F$ is complex differentiable at $z$ with $F'(z) = f(z)$ . Thus $F$ is holomorphic on $B$ and $F' = f$ there.
By holomorphic functions are infinitely differentiable, $F'$ is holomorphic on $B$ . So $f$ is holomorphic on $B$ .
Each point of $U$ lies in such a disc $B$ . So $f$ is complex differentiable at each point of $U$ , and $f$ is holomorphic on $U$ .

Idea

On a disc, integrating $f$ along segments from the center gives a function $F$ . The triangle condition makes the difference $F(z + h) - F(z)$ an integral over the short segment from $z$ to $z + h$ , so continuity of $f$ gives $F' = f$ . The derivative of a holomorphic function is holomorphic, so $f$ is holomorphic.

Reading path

To understand this page from the ground up, read these 352 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof that metric spaces are Hausdorff, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that continuous partial derivatives imply differentiability, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

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Archimedean ordered field definition
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Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
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The reals form an ordered field statement
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Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
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Proof that closed sets contain their limits proof
Continuity at a point of a topological space definition
Proof that open balls are open proof
Compact set definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
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Closed sets contain their limits statement
Open balls are open statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Proof of the intermediate value theorem by the supremum proof
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Proof that metric spaces are Hausdorff proof
Proof of continuity via open sets in metric spaces proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the reverse triangle inequality proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of the limit of a sum of sequences proof
Compactness via open covers statement
Proof of the uniqueness of limits proof
Intermediate value theorem statement
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Metric spaces are Hausdorff statement
Proof that compact subsets of a Hausdorff space are closed proof
Epsilon-delta continuity in metric spaces statement
Reverse triangle inequality statement
Proof of Cantor's intersection theorem proof
Proof that compact subsets of a metric space are bounded proof
Proof that compactness and sequential compactness agree in metric spaces proof
Lebesgue number lemma statement
Proof that a continuous image of a compact set is compact proof
Proof of the triangle inequality for norms proof
Proof of the sequence characterisation of uniform continuity proof
Differentiable functions are continuous statement
Linearity of the integral statement
Limit of a sum of sequences statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
Proof of the Cauchy–Riemann equations proof
Compact subsets of a Hausdorff space are closed statement
Continuity via open sets statement
Cantor's intersection theorem statement
Compact subsets of a metric space are bounded statement
Compactness and sequential compactness agree in metric spaces statement
Epsilon-delta uniform continuity statement
Continuous image of a compact set is compact statement
Proof of the limit of a product of sequences proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Triangle inequality for norms statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Cauchy–Riemann equations statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that Lipschitz maps are uniformly continuous proof
Proof that absolutely convergent series in a Banach space converge proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Geometric series example
Proof of Riemann's criterion for integrability proof
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Decreasing sequence bounded below converges statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bounded linear maps are continuous statement
Comparison test statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Product of continuous functions is continuous statement
Chain rule statement
Ratio test statement
Proof of the Heine–Borel theorem by sequential compactness proof
Additivity of the integral over intervals statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Heine–Borel theorem statement
Proof that power series are holomorphic in the disc of convergence proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the constant multiple rule proof
Proof of the first fundamental theorem of calculus proof
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the estimation lemma proof
Proof of the mean value theorem for integrals proof
Estimation lemma statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that continuous partial derivatives imply differentiability proof
Proof that the integral of a continuous function is an antiderivative proof
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Proof of integration by substitution proof
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the fundamental theorem of contour integration proof
Fundamental theorem of contour integration statement
Proof of Goursat's theorem by repeated subdivision proof
Goursat's theorem statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Cauchy's integral formula statement
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof that holomorphic functions are infinitely differentiable proof
Holomorphic functions are infinitely differentiable statement
Proof of Morera's theorem by a local primitive proof

Where this leads

6 pages build on this one.

Tag	`3ENC7Y5VBE`
URI	`m2://3ENC7Y5VBE`
Kind	proof
Writing	W0 stub
MSC	30E20
Proves	Morera's theorem
Uses	Contour integral, Holomorphic function, Complex differentiability, Integration by substitution, Estimation lemma, Holomorphic functions are infinitely differentiable
Changed	4 November 2026

Proof of Morera's theorem by a local primitive

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