Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities

We develop a theory of specular differentiation on real intervals. The specular derivative is defined by averaging the angles associated with the forward and backward difference quotients and extends classical differentiation. For specularly differentiable and continuous functions, we establish a quasi-mean value theorem and use it to show that continuity of the specular derivative implies $C^1$-regularity. Without assuming continuity, we prove that the discontinuities at which the specular derivative is nonzero form an at most countable set; a nowhere-continuous example with identically zero specular derivative shows that this restriction is sharp. In the second-order theory of specular differentiation, we impose additional conditions on twice specularly differentiable functions to obtain a class that lies strictly between the classes $C^1$ and $C^2$.