A differentiability criterion for continuous functions

We show that, with the exception of the symmetric derivative, each limit of the form lim h → 0 A f ( x + a h ) + B f ( x + b h ) h , ( A + B = 0 , A a + B b = 1 ) , is equivalent to the ordinary derivative, for all continuous functions at x . And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at x , by taking limits of first order difference quotients with two function evaluations.