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?



Harmonic Functions
Yes, the Taylor series works. Actually $C^2$ suffices for the remainder term, although my sophomore calculus book gives the proof using $C^3.$ I get
$$ 4 f(x\_0, y\_0) = 4 f(x\_0, y\_0) + \left( 2 f\_{xx}(x\_0, y\_0) + 2 f\_{yy}(x\_0, y\_0) \right) \delta^2 \; + \; o( \delta^2 ) $$
and
$$ \left( 2 f\_{xx}(x\_0, y\_0) + 2 f\_{yy}(x\_0, y\_0) \right) \delta^2 \; = \; o( \delta^2 ) $$
and 
$$ 2 \left( f\_{xx}(x\_0, y\_0) + f\_{yy}(x\_0, y\_0) \right) \; = \; 0 $$ 


LATER EDIT: unless I am vastly mistaken this argument still works if we put in the caveat $ | \delta | < \Delta = \Delta(x\_0, y\_0), $ that is we only require your equation for small $\delta$ and even say that the allowable size of $\delta$ depends on the position of the center point that I am calling $(x\_0, y\_0).$ But with this change we can build an easy discontinuous example of your relation, take 
$$ f(x\_0, y\_0) = 1, \; \; if \; \; y\_0 > 0, $$
$$ f(x\_0, y\_0) = 0, \; \; if \; \; y\_0 = 0, $$
$$ f(x\_0, y\_0) = -1, \; \; if \; \; y\_0 < 0. $$


Then your relation holds for $ | \delta | < | y\_0 | $ when $y\_0 \neq 0$ and holds for all $\delta$ when $ y\_0 = 0.$


