Differentiable function

A differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

More generally, if x₀ is a point in the domain of a function f, then f is said to be differentiable at x₀ if the derivative f ′(x₀) exists. This means that the graph of f has a non-vertical tangent line at the point (x₀, f(x₀)). The function f may also be called locally linear at x₀, as it can be well approximated by a linear function near this point.

Differentiability of real functions of one variable[edit]

A function $f:U\subset \mathbb {R} \to \mathbb {R}$ , defined on an open set $U$ , is said to be differentiable at $a\in U$ if any of the following equivalent conditions is satisfied:

The derivative $f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}$ exists.
There exists a real number $L$ such that $\lim _{h\to 0}{\frac {f(a+h)-f(a)-Lh}{h}}=0$ . The number $L$ , when it exists, is equal to $f'(a)$ .
There exists a function $g:U\subset \mathbb {R} \to \mathbb {R}$ such that $f(a+h)=f(a)+f'(a)h+g(h)$ and $\lim _{h\to 0}{\frac {g(h)}{h}}=0$ .

Differentiability and continuity[edit]

The 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 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 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, 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 meager 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.

Differentiability classes[edit]

Differentiable functions can be locally approximated by linear functions.

The function

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

with

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

for

x\neq 0

and

f(0)=0

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

A function f is said to be continuously differentiable if the derivative f′(x) 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

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 _{\epsilon \to 0}\left({\frac {\epsilon ^{2}\sin(1/\epsilon )-0}{\epsilon }}\right)=0,

exists. However, for x ≠ 0, differentiation rules imply

f'(x)=2x\sin(1/x)-\cos(1/x)

which has no limit as x → 0. Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem.

Continuously differentiable functions are sometimes said to be of class C¹. A function is of class C² if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class C^k if the first k derivatives f′(x), f′′(x), ..., f^(k)(x) all exist and are continuous. If derivatives f⁽ⁿ⁾ exist for all positive integers n, the function is smooth or equivalently, of class C^∞.

Differentiability in higher dimensions[edit]

A function of several real variables $f : R m \to R n$ is said to be differentiable at a point $x 0$ if there exists a linear map $J : R m \to R n$ such that

\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 exist at $x 0$ , and the linear map $J$ is given by the Jacobian matrix. 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 $x 0$ 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) does not in general guarantee that a function is differentiable at a point. For example, the function $f : R 2 \to R$ defined by

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)={\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[edit]

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 $f:\mathbb {C} \to \mathbb {C}$ is said to be differentiable at $x=a$ when

f'(a)=\lim _{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 $f:\mathbb {C} \to \mathbb {C}$ , that is complex-differentiable at a point $x=a$ is automatically differentiable at that point, when viewed as a function $f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$ . This is because the complex-differentiability implies that

\lim _{h\to 0}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0.

However, a function $f:\mathbb {C} \to \mathbb {C}$ can be differentiable as a multi-variable function, while not being complex-differentiable. For example, $f(z)={\frac {z+{\overline {z}}}{2}}$ is differentiable at every point, viewed as the 2-variable real function $f(x,y)=x$ , but it is not complex-differentiable at any point.

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.