the sufficient condition for differentiability
The Cauchy-Riemann equations are necessary but, on their own, not quite enough to guarantee a complex derivative. The sufficient condition fills the gap: IF the four partial derivatives u_x, u_y, v_x, v_y exist and are CONTINUOUS near a point, AND they satisfy u_x = v_y, u_y = -v_x there, THEN f is complex-differentiable at that point. Continuity of the partials (or equivalently real-differentiability of (u, v)) is the missing ingredient.
Why the extra hypothesis? Real-differentiability means f(z) - f(z_0) is well approximated by a single linear map of the displacement (with error vanishing faster than the step), not merely that directional limits along the axes happen to agree. The Cauchy-Riemann equations then upgrade that linear map into a rotation-and-scaling, i.e. multiplication by a complex number, which is exactly the complex derivative. Continuous partials are a clean, checkable way to know the real linear approximation genuinely exists.
The hypothesis is not pedantry — there are textbook functions where the partials exist and CR holds at a single point yet no complex derivative exists there, because the partials are wildly discontinuous. In practice this almost never bites: any function built from z by addition, multiplication, division (off zeros), and composition has continuous partials automatically, so once CR holds you are safely differentiable.
For f(z) = e^x cos y + i e^x sin y (which is e^z): u = e^x cos y, v = e^x sin y. The partials u_x = e^x cos y = v_y and u_y = -e^x sin y = -v_x are continuous everywhere and CR holds, so e^z is differentiable everywhere with derivative u_x + i v_x = e^z.
Continuous partials plus CR is a complete, practical test for complex differentiability.
CR alone is not sufficient: the Looman-Menchoff theorem weakens 'continuous partials' considerably, but the naive 'CR at a point implies differentiable' is false. Continuity of the partials is the safe, standard hypothesis.