Suppose $f: \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is has continuous partial derivatives and

$$4f(x,y)=f(x+\delta,y+\delta)+f(x-\delta,y+\delta)+f(x-\delta,y-\delta) + f(x+\delta,y-\delta)$$

for all $(x,y)$ in $\mathbb{R}\times\mathbb{R}$ and all $\delta$ in $\mathbb{R}$.

I don't believe that $f$ is necessarly harmonic but I cannot construct a counter-example. Is $f$ harmonic?