The fact that values of the derivative must be the same regardless of the direction of the limit gives rise to this system of equations.
In dimensions greater than two, solutions to this system cannot have isolated singularities or zeroes by Hartogs phenomenon. This system is the simplest example of an auto-Backlund transformation. Exponential integrals of functions that satisfy this system have paths of steepest descent stemming out from saddle points such that the phase remains constant. In fluid mechanics, The stream function and the velocity potential together combine to form a new potential that satisfies this system in potential flow. The fact that values of the derivative must be the same regardless of the direction of the limit gives rise to this system of equations. The real and imaginary parts of functions that satisfy these PDEs also satisfy Laplace's equation separately. For 10 points, name this system of equations which gives the conditions for a function to be complex differentiable and thus analytic.
In dimensions greater than two, solutions to this system cannot have isolated singularities or zeroes by Hartogs phenomenon.
Cauchy-Riemann equations