On the stability of the equation stemming from Lagrange MVT

The stability behaviour of the functional equation F ( y ) − F ( x ) = ( y − x ) f ( ( x + y ) / 2 ) is studied. It is proved that this equation is superstable i.e. if f , F satisfy | F ( y ) − F ( x ) ( y − x ) f ( ( x + y ) / 2 ) | ≤ ε then f satisfies this equation (with some F ). In order to obtain this result the equation h Δ h 2 f ( x ) = 0 is considered and it is proved that also this equation is superstable.