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Suppose a continuous function $u : \mathbb{C} → \mathbb{R}$ has the following property: $u(x + iy) = \frac{1}{4}[(u(x + a + iy) + u(x − a + iy) + u(x + i(y + a)) + u(x + i(y − a)))]$ for all $a\in\mathbb{C}$. Does it imply that u is harmonic?

I am inclined to believe so, but when I true to compute a mean value property for any $z_0\in\mathbb{C}$, I get: $\frac{1}{2\pi}\int_0^{2\pi}u(re^{i\theta}+z_0)d\theta=\frac{1}{2\pi}\sum_{k=0}^3\int_{\frac{k\pi}{4}}^{\frac{(k+1)\pi}{4}} u(re^{i\theta}+z_0)d\theta=\frac{1}{2\pi}\frac{1}{4}\int_{0}^{\frac{\pi}{4}}u(z_0)d\theta=\frac{u(z_0)}{2}\neq u(z_0)$.

Where am I going wrong?

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The idea is good, but you have to split the integration path into four segments of angle $\pi/2$, not $\pi/4$. Also the factor $1/4$ in the third expression should be $4$: $$ \frac{1}{2\pi}\int_0^{2\pi}u(re^{i\theta}+z_0)d\theta=\frac{1}{2\pi}\sum_{k=0}^3\int_{\frac{k\pi}{2}}^{\frac{(k+1)\pi}{2}} u(re^{i\theta}+z_0)d\theta=\frac{1}{2\pi} \int_{0}^{\frac{\pi}{2}}4 u(z_0)d\theta= u(z_0) $$

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