let $f$ be a complex function defined as
$$f(x+iy)=\frac{x}{1+\sqrt{x^2+y^2}}+i\frac{y}{1+\sqrt{x^2+y^2}}$$

Is this $f$ holomorphic?

my first thought is to use Cauchy-Riemann equations, but I am not sure how to use them. Also, I have the feeling that this function is not holomorphic.