Note that the hypothesis of real differentiability at the point is essential and cannot be dispensed with. For example,, 2.14 the function , regarded as a complex function with imaginary part identically zero, has both partial derivatives at , and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.
Note that the hypothesis of real differentiability at the point is essential and cannot be di,spensed with. For example,, 2.14 the function , regarded as a complex function with imaginary part identically zero, has both partial derivatives at , and it moreover satisfies the Cauchy–Riemann equations at that point, :but it is not differentiable in the sense of real functio?ns (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.