the Cauchy-Riemann equations as a necessary condition

This is the easy half of the Cauchy-Riemann story: IF f is complex-differentiable at a point, THEN its real and imaginary parts must satisfy u_x = v_y and u_y = -v_x there. The word 'necessary' means these equations are an unavoidable consequence — they are a test every differentiable function has to pass.

The proof is just the difference quotient evaluated along two directions. Since the limit f'(z_0) exists, any way of approaching z_0 must give the same value. Approaching horizontally (z = z_0 + h with h real, h -> 0) yields f'(z_0) = u_x + i v_x. Approaching vertically (z = z_0 + i k with k real, k -> 0, and the step in the denominator being i k) yields f'(z_0) = (u_y + i v_y)/i = v_y - i u_y. Setting the two expressions equal and matching real and imaginary parts gives u_x = v_y and v_x = -u_y.

Used as a test, the necessary condition is a quick way to PROVE a function is NOT holomorphic: just exhibit a point where CR fails. For instance f(z) = z-bar has u = x, v = -y, so u_x = 1 but v_y = -1; the equations fail everywhere, so z-bar is nowhere differentiable. The condition cannot, by itself, prove a function IS differentiable — that needs the sufficient condition.