Symmetric derivative

In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as The expression under the limit is sometimes called the symmetric difference quotient. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point. If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f(x) = x, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved. For the absolute value function , using the notation for the symmetric derivative, we have at that Hence the symmetric derivative of the absolute value function exists at and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at ). Note that in this example both the left and right derivatives at 0 exist, but they are unequal (one is −1, while the other is +1); their average is 0, as expected. For the function , at we have Again, for this function the symmetric derivative exists at , while its ordinary derivative does not exist at due to discontinuity in the curve there. Furthermore, neither the left nor the right derivative is finite at 0, i.e. this is an essential discontinuity. The Dirichlet function, defined as has a symmetric derivative at every , but is not symmetrically differentiable at any ; i.e. the symmetric derivative exists at rational numbers but not at irrational numbers.