# Semi-differentiability

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Property of a mathematical function

In [calculus](/source/Calculus), the notions of **one-sided differentiability** and **semi-differentiability** of a [real](/source/Real_number)-valued [function](/source/Function_(mathematics)) *f* of a real variable are weaker than [differentiability](/source/Differentiability). Specifically, the function *f* is said to be **right differentiable** at a point *a* if, roughly speaking, a [derivative](/source/Derivative_(mathematics)) can be defined as the function's argument *x* moves to *a* from the right, and **left differentiable** at *a* if the derivative can be defined as *x* moves to *a* from the left.

## One-dimensional case

This function does not have a derivative at the marked point, as the function is not [continuous](/source/Continuous_function) there. However, it has a right derivative at all points, with

          ∂

            +

        f
        (
        a
        )

    {\displaystyle \partial _{+}f(a)}

 constantly equal to 0.

In [mathematics](/source/Mathematics), a **left derivative** and a **right derivative** are [derivatives](/source/Derivative) (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.

### Definitions

Let *f* denote a real-valued function defined on a subset *I* of the real numbers.

If *a* ∈ *I* is a [limit point](/source/Limit_point) of *I* ∩ [*a*,∞) and the [one-sided limit](/source/One-sided_limit)

- ∂ + f ( a ) := lim x → a + x ∈ I f ( x ) − f ( a ) x − a {\displaystyle \partial _{+}f(a):=\lim _{\scriptstyle x\to a^{+} \atop \scriptstyle x\in I}{\frac {f(x)-f(a)}{x-a}}}

exists as a real number, then *f* is called **right differentiable** at *a* and the limit *∂*+*f*(*a*) is called the **right derivative** of *f* at *a*.

If *a* ∈ *I* is a limit point of *I* ∩ (–∞,*a*] and the one-sided limit

- ∂ − f ( a ) := lim x → a − x ∈ I f ( x ) − f ( a ) x − a {\displaystyle \partial _{-}f(a):=\lim _{\scriptstyle x\to a^{-} \atop \scriptstyle x\in I}{\frac {f(x)-f(a)}{x-a}}}

exists as a real number, then *f* is called **left differentiable** at *a* and the limit *∂*–*f*(*a*) is called the **left derivative** of *f* at *a*.

If *a* ∈ *I* is a limit point of *I* ∩ [*a*,∞) and *I* ∩ (–∞,*a*] and if *f* is left and right differentiable at *a*, then *f* is called **semi-differentiable** at *a*.

If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a [symmetric derivative](/source/Symmetric_derivative), which equals the [arithmetic mean](/source/Arithmetic_mean) of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.[1]

### Remarks and examples

- A function is [differentiable](/source/Derivative) at an [interior point](/source/Interior_point) *a* of its [domain](/source/Domain_of_a_function) if and only if it is semi-differentiable at *a* and the left derivative is equal to the right derivative.

- An example of a semi-differentiable function, which is not differentiable, is the [absolute value](/source/Absolute_value) function f ( x ) = | x | {\displaystyle f(x)=|x|} , at *a* = 0. We find easily ∂ − f ( 0 ) = − 1 , ∂ + f ( 0 ) = 1. {\displaystyle \partial _{-}f(0)=-1,\partial _{+}f(0)=1.}

- If a function is semi-differentiable at a point *a*, it implies that it is continuous at *a*.

- The [indicator function](/source/Indicator_function) 1[0,∞) is right differentiable at every real *a*, but discontinuous at zero (note that this indicator function is not left differentiable at zero).

### Application

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](/source/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.

**Theorem**— Let *f* be a real-valued, [continuous function](/source/Continuous_function), defined on an arbitrary [interval](/source/Interval_(mathematics)) *I* of the real line. If *f* is right differentiable at every point *a* ∈ *I*, which is not the [supremum](/source/Supremum) of the interval, and if this right derivative is always zero, then *f* is [constant](/source/Constant_function).

**Proof**

For a [proof by contradiction](/source/Proof_by_contradiction), assume there exist *a* < *b* in *I* such that *f*(*a*) ≠ *f*(*b*). Then

- ε := | f ( b ) − f ( a ) | 2 ( b − a ) > 0. {\displaystyle \varepsilon :={\frac {|f(b)-f(a)|}{2(b-a)}}>0.}

Define *c* as the [infimum](/source/Infimum) of all those *x* in the interval (*a*,*b*] for which the [difference quotient](/source/Difference_quotient) of *f* exceeds *ε* in absolute value, i.e.

- c = inf { x ∈ ( a , b ] ∣ | f ( x ) − f ( a ) | > ε ( x − a ) } . {\displaystyle c=\inf\{\,x\in (a,b]\mid |f(x)-f(a)|>\varepsilon (x-a)\,\}.}

Due to the continuity of *f*, it follows that *c* < *b* and |*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*] with |*f*(*x*) – *f*(*c*)| ≤ *ε*(*x* – *c*) for all *x* in (*c*,*d*]. Hence, by the [triangle inequality](/source/Triangle_inequality),

- | f ( x ) − f ( a ) | ≤ | f ( x ) − f ( c ) | + | f ( c ) − f ( a ) | ≤ ε ( x − a ) {\displaystyle |f(x)-f(a)|\leq |f(x)-f(c)|+|f(c)-f(a)|\leq \varepsilon (x-a)}

for all *x* in [*c*,*d*), which contradicts the definition of *c*.

### Differential operators acting to the left or the right

Another common use is to describe derivatives treated as [binary operators](/source/Binary_operator) in [infix notation](/source/Infix_notation), in which the derivatives is to be applied either to the left or right [operands](/source/Operand). This is useful, for example, when defining generalizations of the [Poisson bracket](/source/Poisson_bracket). For a pair of functions f and g, the left and right derivatives are respectively defined as

- f ∂ x ← g = ∂ f ∂ x ⋅ g {\displaystyle f{\stackrel {\leftarrow }{\partial _{x}}}g={\frac {\partial f}{\partial x}}\cdot g}

- f ∂ x → g = f ⋅ ∂ g ∂ x . {\displaystyle f{\stackrel {\rightarrow }{\partial _{x}}}g=f\cdot {\frac {\partial g}{\partial x}}.}

In [bra–ket notation](/source/Bra%E2%80%93ket_notation), the derivative operator can act on the right operand as the regular derivative or on the left as the negative derivative.[2]

## Higher-dimensional case

This above definition can be generalized to real-valued functions *f* defined on subsets of **R***n* using a weaker version of the [directional derivative](/source/Directional_derivative). 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***n* the limit

- ∂ u f ( a ) = lim h → 0 + f ( a + h u ) − f ( a ) h {\displaystyle \partial _{\mathbf {u} }f(\mathbf {a} )=\lim _{h\to 0^{+}}{\frac {f(\mathbf {a} +h\mathbf {u} )-f(\mathbf {a} )}{h}}}

exists as a real number, with *h* ∈ **R**.

Semi-differentiability is thus weaker than [Gateaux differentiability](/source/Gateaux_derivative), for which one takes in the limit above *h* → 0 without restricting *h* to only positive values.

For example, the function f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)={\sqrt {x^{2}+y^{2}}}} is semi-differentiable at ( 0 , 0 ) {\displaystyle (0,0)} , but not Gateaux differentiable there. Indeed, f ( h x , h y ) = | h | f ( x , y ) and for h ≥ 0 , f ( h x , h y ) = h f ( x , y ) , f ( h x , h y ) / h = f ( x , y ) , {\displaystyle f(hx,hy)=|h|f(x,y){\text{ and for }}h\geq 0,f(hx,hy)=hf(x,y),f(hx,hy)/h=f(x,y),} with a = ( 0 , 0 ) , u = ( x , y ) and ∂ u f ( 0 , 0 ) = f ( x , y ) . {\displaystyle \mathbf {a} =(0,0),\mathbf {u} =(x,y){\text{ and }}\partial _{\mathbf {u} }f(0,0)=f(x,y).}

(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.)

## Properties

- Any [convex function](/source/Convex_function) on a convex [open subset](/source/Open_set) of **R***n* is semi-differentiable.

- While every semi-differentiable function of one variable is continuous; this is no longer true for several variables.

## Generalization

Instead of real-valued functions, one can consider functions taking values in **R***n* or in a [Banach space](/source/Banach_space).

## See also

- [Directional derivative](/source/Directional_derivative)

- [Partial derivative](/source/Partial_derivative)

- [Gradient](/source/Gradient)

- [Gateaux derivative](/source/Gateaux_derivative)

- [Fréchet derivative](/source/Fr%C3%A9chet_derivative)

- [Derivative (generalizations)](/source/Derivative_(generalizations))

- [Phase space formulation § Star product](/source/Phase_space_formulation#Star_product)

- [Dini derivatives](/source/Dini_derivative)

## References

1. **[^](#cite_ref-Mercer2014_1-0)** Peter R. Mercer (2014). *More Calculus of a Single Variable*. Springer. p. 173. [ISBN](/source/ISBN_(identifier)) [978-1-4939-1926-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4939-1926-0).

1. **[^](#cite_ref-2)** Dirac, Paul (1982) [1930]. *The Principles of Quantum Mechanics*. USA: Oxford University Press. [ISBN](/source/ISBN_(identifier)) [978-0198520115](https://en.wikipedia.org/wiki/Special:BookSources/978-0198520115).

- Preda, V.; Chiţescu, I. (1999). "On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case". *J. Optim. Theory Appl*. **100** (2): 417–433. [doi](/source/Doi_(identifier)):[10.1023/A:1021794505701](https://doi.org/10.1023%2FA%3A1021794505701). [S2CID](/source/S2CID_(identifier)) [119868047](https://api.semanticscholar.org/CorpusID:119868047).

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Adapted from the Wikipedia article [Semi-differentiability](https://en.wikipedia.org/wiki/Semi-differentiability) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Semi-differentiability?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.