Semi-differentiability

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In mathematics, a real-valued function f of a real variable is called semi-differentiable at a point x0 if the one-sided limits

\lim_{h\to 0^+}\frac{f(x_0+h)-f(x_0)}{h}

and

\lim_{h\to 0^+}\frac{f(x_0-h)-f(x_0)}{h}

exist.

A function is differentiable at x0 if and only if it is semi-differentiable at x0 and the first of the above limits is the opposite of the second. An example of a semi-differentiable function which is not differentiable is the absolute value at x0 = 0.

This definition can be generalized to functions of several variables. A function f is semi-differentiable at a point x0 in Rn if for any vector u the limit

f^{+}(x_0, u)=\lim_{h\to 0^+}\frac{f(x_0+h\, u)-f(x_0)}{h}

exists.

Semi-differentiability is thus weaker than Gâteaux differentiability, for which one takes in the limit above h\to 0 without restricting h to only positive values.

[edit] Properties

  • Any convex function on an open set is semi-differentiable.
  • Any semi-differentiable function of one variable is continuous; this is no longer true for several variables.

[edit] References

  • Preda, V. and Chiţescu, I. On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl. 100 (1999), no. 2, 417--433.
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