There exist real functions which are continuous at each irrational and dis-

continuous at each rational number. For instance, $\displaystyle f (p/q) = 1/q$ if p/q is in

its lowest terms with q > 0 and with f (irrationals) = 0 is one such. Prove

that there is no function f : R → R which is differentiable at every irrational

and discontinuous at every rational.