Liouville’s theorem
Holomorphy on a proper subset. \(f(z) = z\) on \(\DD\) is bounded and holomorphic and not constant; Liouville’s theorem requires \(f\) to be entire.
Boundedness on \(\RR\) only. \(f(z) = \sin z\) is entire and bounded on \(\RR\), and \(\abs{\sin(iy)} = \sinh\abs y \to \infty\) as \(\abs y\to\infty\).
An entire function with \(\abs{f(z)} \leq C\abs z^n\) for large \(\abs z\) is a polynomial of degree at most \(n\) (Cauchy estimates and Liouville), so \(f(z)=z^n\) is nonconstant under this bound.
The identity principle
Limit point outside the domain. \(f(z) = \sin(\pi/z)\) on \(\CC\smz\) vanishes at the points \(1/k\), which accumulate at \(0\notin\CC\smz\), and \(f\not\equiv 0\).
No limit point. Distinct polynomials of degree at most \(n\) can agree at \(n\) points; a finite set has no limit point.
The maximum and minimum modulus principles
Zeros in the minimum modulus principle. \(f(z) = z\) on \(\DD\) is nonconstant, and \(\abs f\) has its minimum at the interior point \(0\), a zero of \(f\).
Unbounded domains. On the strip \(S\coloneqq\ts{\abs{\Im z}<\pi/2}\), the function \(f(z) = \exp(e^z)\) is holomorphic on \(\overline S\) with \(\abs f = 1\) on \(\bd S\), since \(e^z = \pm i e^x\) there, and \(f(x) = \exp(e^x)\) is unbounded on \(\RR\subseteq S\). By the Phragmén–Lindelöf principle for \(S\), if \(f\) is holomorphic on \(S\), continuous on \(\overline S\), \(\abs f\leq 1\) on \(\bd S\), and \(\abs{f(z)}\leq \exp\qty{Ae^{\alpha\abs{\Re z}}}\) on \(S\) for some \(A>0\) and \(\alpha<1\), then \(\abs f\leq 1\) on \(S\); the function \(\exp(e^z)\) has this growth with \(\alpha=1\).
Singularities
All three types on the same punctured disc. On \(\DD\smz\), \(1\) has a removable singularity, \(1/z\) a pole, and \(e^{1/z}\) an essential singularity at \(0\).
Bounded near an undefined point. \(\sin(z)/z\) is holomorphic and bounded on a punctured disc about \(0\), and by Riemann’s theorem it extends holomorphically with value \(1\).
Singularities outside the classification. \(\sqrt z\) and \(\Log z\) have branch points at \(0\) and no Laurent expansion on any punctured disc about \(0\), so the classification into removable singularities, poles, and essential singularities does not apply. \(1/\sin(\pi/z)\) has a non-isolated singularity at \(0\).
Convergence
Pointwise limits. A pointwise limit of holomorphic functions need not be holomorphic: by Runge’s theorem there are polynomials converging pointwise on \(\CC\) to a discontinuous function. By Montel’s theorem, a sequence of holomorphic functions whose pointwise limit is not holomorphic is not uniformly bounded on compact subsets.
Univalent maps with a constant limit. \(f_n(z) = z/n\) is injective for each \(n\), and \(f_n\to 0\) locally uniformly; Hurwitz’s theorem allows the limit of univalent functions to be constant.
Conformal maps
\(\CC\) and \(\DD\) are not biholomorphic. Any holomorphic map \(\CC\to\DD\) is a bounded entire function, hence constant by Liouville’s theorem; the Riemann mapping theorem requires \(\Omega \neq \CC\).
Nonvanishing derivative without injectivity. \(e^z\) has derivative \(e^z \neq 0\) everywhere and \(e^0 = e^{2\pi i}\); nonvanishing derivative gives only local injectivity.
Injective with vanishing derivative, over \(\RR\). \(x\mapsto x^3\) is injective on \(\RR\), has derivative \(0\) at \(0\), and has inverse \(x^{1/3}\), which is not differentiable at \(0\). An injective holomorphic function has nonvanishing derivative (Injective holomorphic maps have nonvanishing derivative).
Holomorphy
Real differentiable and not holomorphic. \(f(z) = \bar z\) is real differentiable everywhere and fails the Cauchy–Riemann equations; its difference quotient \(\bar h/h\) depends on the direction of \(h\).
Cauchy–Riemann equations at one point. \(f(x+iy) = \sqrt{\abs{xy}}\) has \(u_x = u_y = v_x = v_y = 0\) at \(0\), so the Cauchy–Riemann equations hold there, but along \(y = x\) the difference quotient is \(\abs x/\bigl((1+i)x\bigr)\), which has no limit as \(x\to 0\).
Invertible real Jacobian without conformality. \(f(z) = \bar z\) has invertible real Jacobian at every point and reverses signed angles; a conformal map is holomorphic with nonvanishing derivative.