The only allowable instance of this function over a  nite  eld is the trivial one, which is equal to zero at zero and is equal to one otherwise.
Ostrowski's theorem categorizes the di erent types of this function for the real numbers. The only allowable instance of this function over a  nite  eld is the trivial one, which is equal to zero at zero and is equal to one otherwise. This function must be subadditive, multiplicative, positive-de nite, and nonnegative; it is also idempotent and even over the real numbers. The standard metric on the real line is given by this function of the   di erence between two numbers. This function is not complex di erentiable anywhere, and its derivative at a nonzero point x in the real line is equal to the sign [spell out after saying] of x. For ten points, identify this function which returns x if x is positive and −x if x is negative.
Ostrowski's theorem categorizes the di erent types of this function for the real numbers.
absolute value