I don't know the relationship about complex differentiable and differentiable everywhere, actually.
From WolframMathWorld : Complex Differentiable, The function is complex differentiable when $f(z)$ has derivative, has continuous partial derivative, and satisfies the Riemann Equation. It said that analytic is equivalent with Complex differentiable.
Suppose i have :
$$f(z)=\cos x \cosh y +i \sin x \sinh y$$
The function is indeed continuous since it has no singularity. But it fails on Cauchy-Riemann Equation.
Is that means it's not complex differentiable? Then, how about differentiable everywhere?
Actually the first thing i thought of before i don't know the definition when i heard "differentiable everywhere" was just involving derivative on the complex plane and has no singularity.