# Differentiable function

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Mathematical function whose derivative exists

A differentiable function

In [mathematical analysis](/source/Mathematical_analysis), a [real](/source/Real_number) or [complex](/source/Complex_number) [function](/source/Function_(mathematics)) of a single variable is **differentiable** if its [derivative](/source/Derivative) exists at each point in its [domain](/source/Domain_of_a_function). For real-valued functions of a real variable, the [graph](/source/Graph_of_a_function) of a differentiable function has a non-[vertical](/source/Vertical_tangent) [tangent line](/source/Tangent_line) at each interior point in its domain. A differentiable function is [locally](/source/Neighbourhood_(mathematics)) approximable by a [linear function](/source/Linear_function) at each interior point, and does not contain any [break](/source/Classification_of_discontinuities), angle, or [cusp](/source/Cusp_(singularity)).

If x 0 {\displaystyle x_{0}} is an interior point in the domain of a real function f {\displaystyle f} , then f {\displaystyle f} is said to be *differentiable at* x 0 {\displaystyle x_{0}} if there exists an L ∈ R {\displaystyle L\in \mathbb {R} } such that for all ϵ > 0 {\displaystyle \epsilon >0} , there exists a δ > 0 {\displaystyle \delta >0} such that for all x ∈ ( x 0 − δ , x 0 ) ∪ ( x 0 , x 0 + δ ) {\displaystyle x\in (x_{0}-\delta ,x_{0})\cup (x_{0},x_{0}+\delta )} , f ( x ) − f ( x 0 ) x − x 0 ∈ ( L − ϵ , L + ϵ ) {\displaystyle {\frac {f(x)-f(x_{0})}{x-x_{0}}}\in (L-\epsilon ,L+\epsilon )} . In other words, the graph of f {\displaystyle f} has a non-vertical tangent line at the point ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} . f {\displaystyle f} is said to be differentiable on a subset U ⊆ R {\displaystyle U\subseteq \mathbb {R} } if it is differentiable at every point in U {\displaystyle U} . f {\displaystyle f} is said to be *[continuously differentiable](/source/Continuously_differentiable_function)* if its derivative is also a continuous function over the domain of f {\textstyle f} .

Continuous functions may be nowhere differentiable in their domain, such as the [Weierstrass function](/source/Weierstrass_function). Taking successive antiderivatives of such a function allows one to obtain a function that is differentiable only a finite number of times, the finite number being any positive integer. Given a positive integer k {\displaystyle k} , f {\displaystyle f} is said to be of class C k {\displaystyle C^{k}} if its first k {\displaystyle k} derivatives f ′ , f ′ ′ , … , f ( k ) {\textstyle f^{\prime },f^{\prime \prime },\ldots ,f^{(k)}} exist and are continuous over the domain of f {\displaystyle f} .

For a multivariable function, as shown [here](#Differentiability_in_higher_dimensions), the differentiability of it is something more complex than the existence of the partial derivatives of it.

## Differentiability of real functions of one variable

A function f : U → R {\displaystyle f:U\to \mathbb {R} } , defined on an open set U ⊂ R {\textstyle U\subset \mathbb {R} } , is said to be *differentiable* at a ∈ U {\displaystyle a\in U} if the derivative

- f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h = lim x → a f ( x ) − f ( a ) x − a {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}}

exists. This implies that the function is [continuous](/source/Continuous_function) at a.

This function f is said to be *differentiable* on U if it is differentiable at every point of U. In this case, the derivative of f is thus a function from U into R . {\displaystyle \mathbb {R} .}

A continuous function is not necessarily differentiable, but a differentiable function is necessarily [continuous](/source/Continuous_function) (at every point where it is differentiable) as is shown below (in the section [Differentiability and continuity](#Differentiability_and_continuity)). A function is said to be *continuously differentiable* if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section [Differentiability classes](#Differentiability_classes)).

### Semi-differentiability

Main article: [Semi-differentiability](/source/Semi-differentiability)

The above definition can be extended to define the derivative at [boundary points](/source/Boundary_(topology)) of the domain that also belong to the domain. The derivative of a function f : A → R {\textstyle f:A\to \mathbb {R} } defined on a closed subset A ⊊ R {\textstyle A\subsetneq \mathbb {R} } of the real numbers, evaluated at a boundary point c {\textstyle c} , can be defined as the following one-sided limit, where the argument x {\textstyle x} approaches c {\textstyle c} such that it is always within A {\textstyle A} :

- f ′ ( c ) = lim x → c x ∈ A f ( x ) − f ( c ) x − c . {\displaystyle f'(c)=\lim _{\scriptstyle x\to c \atop \scriptstyle x\in A}{\frac {f(x)-f(c)}{x-c}}.}

For x {\textstyle x} to remain within A {\textstyle A} , which is a subset of the reals, it follows that this limit will be defined as either

- f ′ ( c ) = lim x → c + f ( x ) − f ( c ) x − c or f ′ ( c ) = lim x → c − f ( x ) − f ( c ) x − c . {\displaystyle f'(c)=\lim _{x\to c^{+}}{\frac {f(x)-f(c)}{x-c}}\quad {\text{or}}\quad f'(c)=\lim _{x\to c^{-}}{\frac {f(x)-f(c)}{x-c}}.}

## Differentiability and continuity

See also: [Continuous function](/source/Continuous_function)

The [absolute value](/source/Absolute_value) function is continuous (i.e. it has no gaps). It is differentiable everywhere *except* at the point *x* = 0, where it makes a sharp turn as it crosses the *y*-axis.

A [cusp](/source/Cusp_(singularity)) on the graph of a continuous function. At zero, the function is continuous but not differentiable.

If *f* is differentiable at a point *x*0, then *f* must also be [continuous](/source/Continuous_function) at *x*0. In particular, any differentiable function must be continuous at every point in its domain. *The converse does not hold*: a continuous function need not be differentiable. For example, a function with a bend, [cusp](/source/Cusp_(singularity)), or [vertical tangent](/source/Vertical_tangent) may be continuous, but fails to be differentiable at the location of the anomaly.

Most functions that occur in practice have derivatives at all points or at [almost every](/source/Almost_everywhere) point. However, a result of [Stefan Banach](/source/Stefan_Banach) states that the set of functions that have a derivative at some point is a [meagre set](/source/Meagre_set) in the space of all continuous functions.[1] Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the [Weierstrass function](/source/Weierstrass_function).

## Differentiability classes

Differentiable functions can be locally approximated by linear functions.

The function

        f
        :

          R

        →

          R

    {\displaystyle f:\mathbb {R} \to \mathbb {R} }

 with

        f
        (
        x
        )
        =

          x

            2

        sin
        ⁡

          (

                1
                x

          )

    {\displaystyle f(x)=x^{2}\sin \left({\tfrac {1}{x}}\right)}

 for

        x
        ≠
        0

    {\displaystyle x\neq 0}

 and

        f
        (
        0
        )
        =
        0

    {\displaystyle f(0)=0}

 is differentiable. However, this function is not continuously differentiable.

Main article: [Smoothness](/source/Smoothness)

A function f {\textstyle f} is said to be [continuously differentiable](/source/Continuously_differentiable_function) if the derivative f ′ ( x ) {\textstyle f^{\prime }(x)} exists and is itself a continuous function. Although the derivative of a differentiable function never has a [jump discontinuity](/source/Jump_discontinuity), it is possible for the derivative to have an [essential discontinuity](/source/Classification_of_discontinuities#Essential_discontinuity). For example, the function f ( x ) = { x 2 sin ⁡ ( 1 / x ) if x ≠ 0 0 if x = 0 {\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}} is differentiable at 0, since f ′ ( 0 ) = lim ε → 0 ( ε 2 sin ⁡ ( 1 / ε ) − 0 ε ) = 0 {\displaystyle f'(0)=\lim _{\varepsilon \to 0}\left({\frac {\varepsilon ^{2}\sin(1/\varepsilon )-0}{\varepsilon }}\right)=0} exists. However, for x ≠ 0 , {\displaystyle x\neq 0,} [differentiation rules](/source/Differentiation_rules) imply f ′ ( x ) = 2 x sin ⁡ ( 1 / x ) − cos ⁡ ( 1 / x ) , {\displaystyle f'(x)=2x\sin(1/x)-\cos(1/x)\;,} which has no limit as x → 0. {\displaystyle x\to 0.} Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless, [Darboux's theorem](/source/Darboux's_theorem_(analysis)) implies that the derivative of any function satisfies the conclusion of the [intermediate value theorem](/source/Intermediate_value_theorem).

Similarly to how [continuous functions](/source/Continuous_function) are said to be of *class C 0 , {\displaystyle C^{0},}* continuously differentiable functions are sometimes said to be of *class C 1 {\displaystyle C^{1}}*. A function is of *class C 2 {\displaystyle C^{2}}* if the first and [second derivative](/source/Second_derivative) of the function both exist and are continuous. More generally, a function is said to be of *class C k {\displaystyle C^{k}}* if the first k {\displaystyle k} derivatives f ′ ( x ) , f ′ ′ ( x ) , … , f ( k ) ( x ) {\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)} all exist and are continuous. If derivatives f ( n ) {\displaystyle f^{(n)}} exist for all positive integers n , {\textstyle n,} the function is [smooth](/source/Smooth_function) or equivalently, of *class C ∞ . {\displaystyle C^{\infty }.}*

## Differentiability in higher dimensions

See also: [Multivariable calculus](/source/Multivariable_calculus) and [Smoothness § Multivariate differentiability classes](/source/Smoothness#Multivariate_differentiability_classes)

A [function of several real variables](/source/Function_of_several_real_variables) **f**: **R***m* → **R***n* is said to be differentiable at a point **x**0 if there exists a [linear map](/source/Linear_map) **J**: **R***m* → **R***n* such that

- lim h → 0 ‖ f ( x 0 + h ) − f ( x 0 ) − J ( h ) ‖ R n ‖ h ‖ R m = 0. {\displaystyle \lim _{\mathbf {h} \to \mathbf {0} }{\frac {\|\mathbf {f} (\mathbf {x_{0}} +\mathbf {h} )-\mathbf {f} (\mathbf {x_{0}} )-\mathbf {J} \mathbf {(h)} \|_{\mathbf {R} ^{n}}}{\|\mathbf {h} \|_{\mathbf {R} ^{m}}}}=0.}

If a function is differentiable at **x**0, then all of the [partial derivatives](/source/Partial_derivative) exist at **x**0, and the linear map **J** is given by the [Jacobian matrix](/source/Jacobian_matrix), an *n* × *m* matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the [fundamental increment lemma](/source/Fundamental_increment_lemma) found in single-variable calculus.

If all the partial derivatives of a function exist in a [neighborhood](/source/Neighbourhood_(mathematics)) of a point **x**0 and are continuous at the point **x**0, then the function is differentiable at that point **x**0.

However, the existence of the partial derivatives (or even of all the [directional derivatives](/source/Directional_derivative)) does not guarantee that a function is differentiable at a point. For example, the function *f*: **R**2 → **R** defined by

- f ( x , y ) = { x if y ≠ x 2 0 if y = x 2 {\displaystyle f(x,y)={\begin{cases}x&{\text{if }}y\neq x^{2}\\0&{\text{if }}y=x^{2}\end{cases}}}

is not differentiable at (0, 0), but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function

- f ( x , y ) = { y 3 / ( x 2 + y 2 ) if ( x , y ) ≠ ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) {\displaystyle f(x,y)={\begin{cases}y^{3}/(x^{2}+y^{2})&{\text{if }}(x,y)\neq (0,0)\\0&{\text{if }}(x,y)=(0,0)\end{cases}}}

is not differentiable at (0, 0), but again all of the partial derivatives and directional derivatives exist.

## Differentiability in complex analysis

Main article: [Holomorphic function](/source/Holomorphic_function)

In [complex analysis](/source/Complex_analysis), complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing [complex numbers](/source/Complex_number). So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when

- f ′ ( a ) = lim h → 0 h ∈ C f ( a + h ) − f ( a ) h . {\displaystyle f'(a)=\lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(a+h)-f(a)}{h}}.}

Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } , that is complex-differentiable at a point x = a {\textstyle x=a} is automatically differentiable at that point, when viewed as a function f : R 2 → R 2 {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} . This is because the complex-differentiability implies that

- lim h → 0 h ∈ C | f ( a + h ) − f ( a ) − f ′ ( a ) h | | h | = 0. {\displaystyle \lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0.}

However, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } can be differentiable as a multi-variable function, while not being complex-differentiable. For example, f ( z ) = z + z ¯ 2 {\displaystyle f(z)={\frac {z+{\overline {z}}}{2}}} is differentiable at every point, viewed as the 2-variable [real function](/source/Real-valued_function) f ( x , y ) = x {\displaystyle f(x,y)=x} , but it is not complex-differentiable at any point because the limit lim h → 0 h + h ¯ 2 h {\textstyle \lim _{h\to 0}{\frac {h+{\bar {h}}}{2h}}} gives different values for different approaches to 0.

Any function that is complex-differentiable in a neighborhood of a point is called [holomorphic](/source/Holomorphic_function) at that point. Such a function is necessarily infinitely differentiable, and in fact [analytic](/source/Analytic_function).

## Differentiable functions on manifolds

See also: [Differentiable manifold § Differentiable functions](/source/Differentiable_manifold#Differentiable_functions)

If *M* is a [differentiable manifold](/source/Differentiable_manifold), a real or complex-valued function *f* on *M* is said to be differentiable at a point *p* if it is differentiable with respect to some (or any) coordinate chart defined around *p*. If *M* and *N* are differentiable manifolds, a function *f*: *M* → *N* is said to be differentiable at a point *p* if it is differentiable with respect to some (or any) coordinate charts defined around *p* and *f*(*p*).

## See also

- [Generalizations of the derivative](/source/Generalizations_of_the_derivative)

- [Semi-differentiability](/source/Semi-differentiability)

- [Differentiable programming](/source/Differentiable_programming)

## References

1. **[^](#cite_ref-1)** Banach, S. (1931). ["Über die Baire'sche Kategorie gewisser Funktionenmengen"](https://doi.org/10.4064%2Fsm-3-1-174-179). *[Studia Math.](/source/Studia_Mathematica)* **3** (1): 174–179. [doi](/source/Doi_(identifier)):[10.4064/sm-3-1-174-179](https://doi.org/10.4064%2Fsm-3-1-174-179).. Cited by Hewitt, E; Stromberg, K (1963). *Real and abstract analysis*. Springer-Verlag. Theorem 17.8.

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