Differentiable function

A differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.

If Template:Math is an interior point in the domain of a function Template:Mvar, then Template:Mvar is said to be differentiable at Template:Math if the derivative <math>f'(x_0)</math> exists. In other words, the graph of Template:Mvar has a non-vertical tangent line at the point Template:Math. Template:Mvar is said to be differentiable on Template:Mvar if it is differentiable at every point of Template:Mvar. Template:Mvar is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function <math display="inline">f</math>. Generally speaking, Template:Mvar is said to be of class Template:Em if its first <math>k</math> derivatives <math display="inline">f^{\prime}(x), f^{\prime\prime}(x), \ldots, f^{(k)}(x)</math> exist and are continuous over the domain of the function <math display="inline">f</math>.

For a multivariable function, as shown here, the differentiability of it is something more complex than the existence of the partial derivatives of it.

Differentiability of real functions of one variableEdit

A function <math>f:U\to\mathbb{R}</math>, defined on an open set <math display="inline">U\subset\mathbb{R}</math>, is said to be differentiable at <math>a\in U</math> if the derivative

exists. This implies that the function is continuous at Template:Mvar.

This function Template:Mvar is said to be differentiable on Template:Mvar if it is differentiable at every point of Template:Mvar. In this case, the derivative of Template:Mvar is thus a function from Template:Mvar into <math>\mathbb R.</math>

A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable) as is shown below (in the section 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).

Semi-differentiabilityEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The above definition can be extended to define the derivative at boundary points. The derivative of a function <math display="inline">f:A\to \mathbb{R}</math> defined on a closed subset <math display="inline">A\subsetneq \mathbb{R}</math> of the real numbers, evaluated at a boundary point <math display="inline">c</math>, can be defined as the following one-sided limit, where the argument <math display="inline">x</math> approaches <math display="inline">c</math> such that it is always within <math display="inline">A</math>:

<math>f'(c)=\lim_Template:\scriptstyle x\to c\atop\scriptstyle x\in A\frac{f(x)-f(c)}{x-c}.</math>

For <math display="inline">x</math> to remain within <math display="inline">A</math>, which is a subset of the reals, it follows that this limit will be defined as either

Differentiability and continuityEdit

Template:See also

File:Absolute value.svg

The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point Template:Math = 0, where it makes a sharp turn as it crosses the Template:Math-axis.

File:Cusp at (0,0.5).svg

A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable.

If Template:Math is differentiable at a point Template:Math, then Template:Math must also be continuous at Template:Math. 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, or 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 point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meagre set in the space of all continuous functions.<ref>Template:Cite journal. Cited by Template:Cite book</ref> 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.

Differentiability classesEdit

File:Approximation of cos with linear functions without numbers.svg

Differentiable functions can be locally approximated by linear functions.

File:The function x^2*sin(1 over x).svg

The function <math>f : \R \to \R</math> with <math>f(x) = x^2\sin\left(\tfrac 1x\right)</math> for <math>x \neq 0</math> and <math>f(0) = 0</math> is differentiable. However, this function is not continuously differentiable.

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A function <math display="inline">f</math> is said to be Template:Em if the derivative <math display="inline">f^{\prime}(x)</math> exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function <math display="block">f(x) \;=\; \begin{cases} x^2 \sin(1/x) & \text{ if }x \neq 0 \\ 0 & \text{ if } x = 0\end{cases}</math> is differentiable at 0, since <math display="block">f'(0) = \lim_{\varepsilon \to 0} \left(\frac{\varepsilon^2\sin(1/\varepsilon)-0}{\varepsilon}\right) = 0</math> exists. However, for <math>x \neq 0,</math> differentiation rules imply <math display="block">f'(x) = 2x\sin(1/x) - \cos(1/x)\;,</math> which has no limit as <math>x \to 0.</math> 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 implies that the derivative of any function satisfies the conclusion of the intermediate value theorem.

Similarly to how continuous functions are said to be of Template:Em continuously differentiable functions are sometimes said to be of Template:Em. A function is of Template:Em if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of Template:Em if the first <math>k</math> derivatives <math display="inline">f^{\prime}(x), f^{\prime\prime}(x), \ldots, f^{(k)}(x)</math> all exist and are continuous. If derivatives <math>f^{(n)}</math> exist for all positive integers <math display="inline">n,</math> the function is smooth or equivalently, of Template:Em

Differentiability in higher dimensionsEdit

A function of several real variables Template:Math is said to be differentiable at a point Template:Math if there exists a linear map Template:Math such that

<math>\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.</math>

If a function is differentiable at Template:Math, then all of the partial derivatives exist at Template:Math, and the linear map Template:Math is given by the Jacobian matrix, an n × m matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus.

If all the partial derivatives of a function exist in a neighborhood of a point Template:Math and are continuous at the point Template:Math, then the function is differentiable at that point Template:Math.

However, the existence of the partial derivatives (or even of all the directional derivatives) does not guarantee that a function is differentiable at a point. For example, the function Template:Math defined by

<math display="block">f(x,y) = \begin{cases}x & \text{if }y \ne x^2 \\ 0 & \text{if }y = x^2\end{cases}</math>

is not differentiable at Template:Math, but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function

<math>f(x,y) = \begin{cases}y^3/(x^2+y^2) & \text{if }(x,y) \ne (0,0) \\ 0 & \text{if }(x,y) = (0,0)\end{cases}</math>

is not differentiable at Template:Math, but again all of the partial derivatives and directional derivatives exist.

Template:See also

Differentiability in complex analysisEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In 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. So, a function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math> is said to be differentiable at <math display="inline">x=a</math> when

<math>f'(a)=\lim_{\underset{h\in\mathbb C}{h\to 0}}\frac{f(a+h)-f(a)}{h}.</math>

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

<math>\lim_{\underset{h\in\mathbb C}{h\to 0}}\frac{|f(a+h)-f(a)-f'(a)h|}{|h|}=0.</math>

However, a function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math> can be differentiable as a multi-variable function, while not being complex-differentiable. For example, <math>f(z)=\frac{z+\overline{z}}{2}</math> is differentiable at every point, viewed as the 2-variable real function <math>f(x,y)=x</math>, but it is not complex-differentiable at any point because the limit <math display="inline">\lim_{h\to 0}\frac{h+\bar h}{2h}</math> gives different values for different approaches to 0.

Any function that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Such a function is necessarily infinitely differentiable, and in fact analytic.