# Give a subharmonic function that is not real analytic

Is there a function $$f(x)$$ defined on an open set of $$\mathbb{R}^{2}$$ ($$k\geq2$$) such that

1) $$f$$ is $$C^{2}$$ smooth,

2) $$f$$ is subharmonic, i.e. the laplacian $$\Delta f$$ of $$f$$ is positive,

3) $$f$$ is not real analytic?

Define $$f(x,y)= |x|^3 + x^2.$$ Then $$f$$ is $$C^2$$ and $$\Delta f(x,y) = 6|x|+2.$$ If $$f$$ were real analytic, then its Laplacian would be real analytic. But clearly this fails.