Complex analysis can feel like a miracle the first time you see it done well. A short argument about complex differentiability suddenly implies an integral formula, then bounds, then rigidity, then that functions which look unrelated must in fact agree everywhere. That speed is inspiring, but it can also hide the real lesson: complex differentiability is not a mild smoothness assumption. It is a structural constraint so strong that a single missing hypothesis can make the entire machine collapse.
A single counterexample can make that constraint vivid.
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The trap: Cauchy–Riemann at a point is not enough
A typical first encounter goes like this.
You define complex differentiability at a point:
- A function $f : \mathbb C \to \mathbb C$ is complex differentiable at $z_0$ if the limit
exists as a complex number.
Then you write $f(x+iy)=u(x,y)+iv(x,y)$ with real-valued $u,v$ and derive the Cauchy–Riemann equations:
as a necessary condition for complex differentiability when the relevant partial derivatives exist.
At this point, it is extremely tempting to think:
- If a function satisfies Cauchy–Riemann at $z_0$, then it should be complex differentiable at $z_0$.
That statement is false.
The right statement is more subtle:
- If $u$ and $v$ have continuous first partial derivatives in a neighborhood of $z_0$ and satisfy Cauchy–Riemann there, then $f$ is holomorphic (complex differentiable in a neighborhood), hence complex differentiable at $z_0$.
- At a single point, or without continuity of partials, Cauchy–Riemann can hold and still fail to guarantee complex differentiability.
The following concrete function makes the gap unmistakable.
The counterexample
Define $f : \mathbb C \to \mathbb C$ by setting $z=x+iy$ and
This is a perfectly explicit formula, and it is built to do two things:
- make the Cauchy–Riemann equations hold at the origin,
- while still breaking complex differentiability at the origin.
Step 1: compute partial derivatives at the origin
We first compute $u(x,y)=\dfrac{x^3}{x^2+y^2}$ and $v(x,y)=\dfrac{y^3}{x^2+y^2}$ for $(x,y)\neq(0,0)$, and $u(0,0)=v(0,0)=0$.
Compute $u_x(0,0)$ using the definition of partial derivative:
When $y=0$, we have $u(h,0)=\dfrac{h^3}{h^2}=h$. So
Compute $u_y(0,0)$:
When $x=0$, we have $u(0,h)=0$. Hence
Similarly for $v$:
But $v(h,0)=0$. So $v_x(0,0)=0$.
And
When $x=0$, $v(0,h)=\dfrac{h^3}{h^2}=h$. Hence $v_y(0,0)=1$.
So at the origin:
- $u_x(0,0)=1$
- $u_y(0,0)=0$
- $v_x(0,0)=0$
- $v_y(0,0)=1$
Step 2: check Cauchy–Riemann at the origin
The Cauchy–Riemann equations at $(0,0)$ require:
- $u_x(0,0)=v_y(0,0)$, which is $1=1$,
- $u_y(0,0)=-v_x(0,0)$, which is $0=0$.
So Cauchy–Riemann holds at the origin.
If you have absorbed the wrong lesson, you might now conclude that $f$ is complex differentiable at $0$. It is not.
Step 3: test the complex difference quotient
Complex differentiability at 0 means the limit
must exist and be the same along every path \to 0.
We will examine two paths.
Along the real axis
Take $z=x$ with $y=0$. For $x\neq 0$,
So
Along this path, the limit is 1.
Along the diagonal $y=x$
Now take $z=x+ix=x(1+i)$, so $y=x$. For $x\neq 0$,
So
Along this path, the limit is $\tfrac12$.
The same limit cannot be both 1 and $\tfrac12$. Therefore the complex derivative at 0 does not exist, and **$f$ is not complex differentiable at the origin**, despite satisfying Cauchy–Riemann at that point.
That is the counterexample.
What the counterexample is really teaching
This single construction forces you to internalize several truths that are easy to say and hard to feel.
Complex differentiability is a neighborhood property in disguise
In real calculus, differentiability at a point is often treated as local: you inspect behavior near that point, but many pathologies are compatible with having a derivative at a single point.
In complex analysis, the definition is still pointwise, but the consequences are inherently global and rigid. The reason is that complex differentiability packages information in every direction simultaneously. When it holds on an open set, it forces integral identities and strong regularity.
The counterexample shows that if you weaken the hypotheses too much, the structure does not partially survive. It fails outright.
Cauchy–Riemann is necessary, but without regularity it is too weak
Why did Cauchy–Riemann hold at 0 in this example? Because the partial derivatives at 0 are computed using one-dimensional limits, and the function was tuned so those one-dimensional limits behave.
But the complex difference quotient probes two-dimensional behavior. It demands that the function align in a way that is consistent across all approaches. The counterexample violates that consistency.
In practical terms, continuity of partial derivatives (or other regularity assumptions) is not a technical luxury. It is what prevents a function from being engineered to fool the pointwise Cauchy–Riemann test.
Holomorphic functions are dramatically more rigid than C¹ real functions
A holomorphic function is infinitely differentiable and equal to its power series locally. Those are not extra assumptions. They are forced consequences of Cauchy’s integral formula.
The counterexample is not holomorphic, and it does not even have a complex derivative at 0. It sits outside that world even though, at one point, it pretends to satisfy a holomorphic signature.
That contrast is the real moral: holomorphic is not “real differentiable plus a little more.”
How to use this lesson when reading or writing proofs
The most common failure mode for students in complex analysis is not algebraic manipulation. It is carrying over intuition from real analysis about how much a condition at a point should buy you.
Here are concrete habits this counterexample trains.
- When a theorem claims holomorphicity from Cauchy–Riemann, immediately check the hypothesis on regularity (continuous partials, or distributional hypotheses, or Morera-type conditions).
- When you see a claim about a derivative at a point, ask whether the proof uses an integral formula or estimates that secretly require holomorphicity on a neighborhood.
- When you attempt to prove a complex limit exists, test multiple paths early, before committing \to a computation.
A small upgrade: why continuity of partial derivatives fixes it
It is worth seeing, at least conceptually, why the standard Cauchy–Riemann theorem includes continuity of partial derivatives.
Suppose $u,v$ have continuous first partial derivatives in a neighborhood of $z_0$ and satisfy Cauchy–Riemann there. Then the differential of $f$ as a map $\mathbb R^2\to\mathbb R^2$ is represented by the Jacobian matrix
Cauchy–Riemann forces this matrix to have the special form
with $a=u_x=v_y$ and $b=v_x=-u_y$. That is exactly the real-linear map corresponding to multiplication by the complex number $a+ib$. Continuity of partials ensures this linear approximation controls the function uniformly in a neighborhood, which is what makes the complex difference quotient converge.
In the counterexample, the partial derivatives exist at 0 but the behavior of the function near 0 is too irregular for that linear approximation to dominate uniformly. The Jacobian at 0 does not control the function in a neighborhood, so complex differentiability fails.
The point of learning complex analysis
This is not merely a technical caution. It clarifies what the subject is actually about.
Complex analysis is about a class of functions defined by a single, stringent local constraint, and then extracting everything that constraint forces:
- integral identities,
- analytic continuation,
- maximum principles,
- conformal geometry,
- residue calculus,
- rigidity phenomena.
To operate in that world, you need to respect the boundary between:
- conditions that are genuinely strong enough to enter the holomorphic regime,
- and conditions that merely imitate it at a point.
This one counterexample draws that boundary in ink.
If you keep it in mind, many proofs become easier because you stop trying to push weak hypotheses through holomorphic machinery. You learn to either strengthen the hypothesis or switch tools.
That is what a good counterexample does: it does not just refute a statement. It teaches you what the subject will and will not allow.
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