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| In [[calculus]], a branch of [[mathematics]], the notions of '''one-sided differentiability''' and '''semi-differentiability''' of a [[real number|real]]-valued [[function (mathematics)|function]] ''f'' of a real variable are weaker than [[differentiability]].
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| ==One-dimensional case==
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| ===Definitions===
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| Let ''f'' denote a real-valued function defined on a subset ''I'' of the real numbers.
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| If ''a'' ∈ ''I'' is a [[limit point]] of ''I'' ∩ <nowiki>[</nowiki>''a'',∞) and the [[one-sided limit]]
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| :<math>\partial_+f(a):=\lim_{{\scriptstyle x\to a+\atop\scriptstyle x\in I}}\frac{f(x)-f(a)}{x-a}</math>
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| exists as a real number, then ''f'' is called '''right differentiable''' at ''a'' and the limit ∂<sub>+</sub>''f''(''a'') is called the '''right derivative''' of ''f'' at ''a''.
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| If ''a'' ∈ ''I'' is a limit point of ''I'' ∩ (–∞,''a''<nowiki>]</nowiki> and the one-sided limit
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| :<math>\partial_-f(a):=\lim_{{\scriptstyle x\to a-\atop\scriptstyle x\in I}}\frac{f(x)-f(a)}{x-a}</math>
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| exists as a real number, then ''f'' is called '''left differentiable''' at ''a'' and the limit ∂<sub>–</sub>''f''(''a'') is called the '''left derivative''' of ''f'' at ''a''.
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| If ''a'' ∈ ''I'' is a limit point of ''I'' ∩ <nowiki>[</nowiki>''a'',∞) and ''I'' ∩ (–∞,''a''<nowiki>]</nowiki> and if ''f'' is left and right differentiable at ''a'', then ''f'' is called '''semi-differentiable''' at ''a''.
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| ===Remarks and examples===
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| * A function is [[derivative|differentiable]] at an [[interior point]] ''a'' of its [[domain (mathematics)|domain]] [[if and only if]] it is semi-differentiable at ''a'' and the left derivative is equal to the right derivative.
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| * An example of a semi-differentiable function, which is not differentiable, is the [[absolute value]] at ''a'' = 0.
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| * A function, which is semi-differentiable at a point ''a'', is also continuous at ''a''.
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| * The [[indicator function]] 1<sub><nowiki>[</nowiki>0,∞)</sub> is right differentiable at every real ''a'', but discontinuous at zero (note that this indicator function is not left differentiable at zero).
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| ===Application===
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| If a real-valued, differentiable function ''f'', defined on an interval ''I'' of the real line, has zero derivative everywhere, then it is constant, as an application of the [[mean value theorem]] shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of ''f''. The version for right differentiable functions is given below, the version for left differentiable functions is analogous.
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| '''Theorem:''' Let ''f'' be a real-valued, [[continuous function]], defined on an arbitrary [[interval (mathematics)|interval]] ''I'' of the real line. If ''f'' is right differentiable at every point ''a'' ∈ ''I'', which is not the [[supremum]] of the interval, and if this right derivative is always zero, then ''f'' is [[constant function|constant]].
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| '''Proof:''' For a [[proof by contradiction]], assume there exist ''a'' < ''b'' in ''I'' such that ''f''(''a'') ≠ ''f''(''b''). Then
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| :<math>\varepsilon:=\frac{|f(b)-f(a)|}{2(b-a)}>0.</math>
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| Define ''c'' as the [[infimum]] of all those ''x'' in the interval (''a'',''b''<nowiki>]</nowiki> for which the [[difference quotient]] of ''f'' exceeds ''ε'' in absolute value, i.e.
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| :<math>c=\inf\{\,x\in(a,b]\mid |f(x)-f(a)|>\varepsilon(x-a)\,\}.</math>
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| Due to the continuity of ''f'', it follows that ''c'' < ''b'' and
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| |''f''(''c'') – ''f''(''a'')| = ''ε''(''c'' – ''a''). At ''c'' the right derivative of ''f'' is zero by assumption, hence there exists ''d'' in the interval (''c'',''b''<nowiki>]</nowiki> with |''f''(''x'') – ''f''(''c'')| ≤ ''ε''(''x'' – ''c'') for all ''x'' in (''c'',''d''<nowiki>]</nowiki>. Hence, by the [[triangle inequality]],
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| :<math>|f(x)-f(a)|\le|f(x)-f(c)|+|f(c)-f(a)|\le\varepsilon(x-a)</math> | |
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| for all ''x'' in <nowiki>[</nowiki>''c'',''d''<nowiki>]</nowiki>, which contradicts the definition of ''c''.
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| ==Higher-dimensional case==
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| This above definition can be generalized to real-valued functions ''f'' defined on subsets of '''R'''<sup>''n''</sup>. Let ''a'' be an interior point of the domain of ''f''. Then ''f'' is called ''semi-differentiable'' at the point ''a'' if for every direction ''u'' ∈ '''R'''<sup>''n''</sup> the limit
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| :<math>\partial_uf(a)=\lim_{h\to 0^+}\frac{f(a+h\, u)-f(a)}{h}</math>
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| exists as a real number.
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| Semi-differentiability is thus weaker than [[Gâteaux derivative|Gâteaux differentiability]], for which one takes in the limit above ''h'' → 0 without restricting ''h'' to only positive values.
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| (Note that this generalization is not equivalent to the original definition for ''n = 1'' since the concept of one-sided limit points is replaced with the stronger concept of interior points.)
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| ==Properties==
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| * Any [[convex function]] on a convex [[open set|open subset]] of '''R'''<sup>''n''</sup> is semi-differentiable.
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| * While every semi-differentiable function of one variable is continuous; this is no longer true for several variables.
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| ==Generalization==
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| Instead of real-valued functions, one can consider functions taking values in '''R'''<sup>''n''</sup> or in a [[Banach space]].
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| ==See also==
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| * [[Derivative]]
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| * [[Directional derivative]]
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| * [[Partial derivative]]
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| * [[Gradient]]
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| * [[Gâteaux derivative]]
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| * [[Fréchet derivative]]
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| * [[Derivative (generalizations)]]
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| ==References==
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| * 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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| [[Category:Real analysis]]
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| [[Category:Differential calculus]]
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| [[Category:Articles containing proofs]]
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The title of the writer is Figures. The factor she adores most is body building and now she is trying to earn cash with it. Hiring is her working day job now and she will not alter it anytime soon. South Dakota is exactly where me and my spouse reside.
My blog post ... http://drupal.12thirty4.com/gmaps/node/4991