The function f(z) = |z|² can be shown to be complex differentiable at z = 0. A student concludes it must therefore be holomorphic on a neighborhood of 0. What is wrong with this reasoning?
AThe function is not continuous at z = 0, so differentiability cannot be established there
BHolomorphic requires complex differentiability at every point in an open domain, not just at one isolated point — |z|² fails complex differentiability everywhere except z = 0
CReal-valued functions of a complex variable are never complex differentiable
DThe student is correct — differentiability at one point is sufficient to establish holomorphicity in a neighborhood
Holomorphic on a domain means complex differentiable at every point of that domain — a global condition on an open set. A function can be complex differentiable at an isolated point without being holomorphic anywhere. The Cauchy-Riemann equations confirm that |z|² = x² + y² satisfies them only at the origin. Option D represents the misconception that confuses pointwise differentiability with holomorphicity.
Question 2 Multiple Choice
In real analysis, a function can be differentiable exactly once — differentiable but not twice differentiable. What is the analogous situation for holomorphic functions?
AThe same situation occurs — a function can be complex differentiable exactly once on a domain
BThere is no such situation: complex differentiability on a domain automatically implies the function is infinitely differentiable
CComplex differentiability is weaker than real differentiability, so 'differentiable once but not twice' functions are more common
DOnly polynomial functions can be holomorphic, and polynomials are always infinitely differentiable
This is one of the most striking differences between real and complex analysis. Once f is complex differentiable on an open domain (holomorphic), all higher derivatives f', f'', f''', ... automatically exist and are themselves holomorphic. The real hierarchy of differentiability classes (C¹ ⊂ C² ⊂ ... ⊂ C∞ ⊂ analytic) collapses in complex analysis: complex differentiability (C¹) immediately implies analyticity. This rigidity has no counterpart in real calculus.
Question 3 True / False
A holomorphic function on a connected domain is completely determined by its values on any open subset of that domain.
TTrue
FFalse
Answer: True
This is the identity theorem, a consequence of holomorphic functions equaling their Taylor series. If two holomorphic functions agree on any open set (or even on a sequence of points converging to a limit point), they must agree everywhere on the connected domain. This global determination from local data has no analogue for smooth real functions, which can be modified locally without affecting values elsewhere — and it is one of the 'rigid' properties that makes holomorphic functions special.
Question 4 True / False
Nearly every differentiable function of a real variable, extended to the complex plane by ignoring the imaginary part — setting f(x + iy) = g(x) — is holomorphic.
TTrue
FFalse
Answer: False
If f(x+iy) = g(x) for a real function g, then u(x,y) = g(x) and v(x,y) = 0. The Cauchy-Riemann equations require ∂u/∂x = ∂v/∂y, which gives g'(x) = 0 for all x — meaning g must be constant. No nonconstant function that depends only on the real part can be holomorphic. Real differentiability and complex differentiability are fundamentally different requirements, and the extension of a real function is almost never holomorphic.
Question 5 Short Answer
Why does complex differentiability impose far stronger constraints than real differentiability, and what is the key geometric reason?
Think about your answer, then reveal below.
Model answer: In real analysis, differentiability at x₀ requires the limit (f(x₀+h)−f(x₀))/h to exist as h approaches 0 from only two directions: left and right. In complex analysis, h is a complex number and can approach 0 from infinitely many directions — every angle in the complex plane. The limit must be the same value regardless of approach direction. This requirement forces the real and imaginary parts of f to satisfy the Cauchy-Riemann equations, which in turn implies that f is not merely once differentiable but infinitely differentiable and equal to its Taylor series.
The 'infinitely many approach directions' constraint is what collapses the differentiability hierarchy. A function passing this test at every point in a domain must be extraordinarily well-behaved — which is why holomorphic functions are so rigid, predictable, and analytically tractable compared to their real counterparts.