If $f = u + iv$ is differentiable at a point z, show that the first-order partials of $u$ and $v$ exist at $z.$ In connection with the discussion: How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$
I would like to ask whether the following statement is true: 

If $f = u + iv$ is differentiable at a point $z$, show that the first-order partials of $u$ and $v$ exist at $z.$


It is a classical result (from any multivariable calculus course) that real (hence complex) differentiability implies the existence of directional derivatives in any directions and in particular the existence of partial derivatives. 
The converse is false. For instance, $f(x,y)=\frac{xy}{x^2+y^2}$ has directional derivatives in any direction but is not differentiable.
