Concept
Differentiable function
Overview
In mathematical analysis, a real or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is locally approximable by a linear function at each interior point, and does not contain any break, angle, or cusp.
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 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. 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, the differentiability of it is something more complex than the existence of the partial derivatives of it. However, the essence is the same; a function that is locally approximable by a linear map at
X
0
{\displaystyle \mathbf {X_{0}} }
is said to be differentiable at this point.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
In this employment the algebraic letters are called the ‘arguments’ of the function, or sometimes they are called the ‘variables.’ Then, for instance, if an angle is represented by an algebraical letter, as standing for its numerical measure in terms of a given unit, Trigonometry is absorbed into this new algebra. Algebra thus develops into the general science of analysis in which we consider the properties of…
Alfred North Whitehead, Science and the modern world
States what Differentiable function is.
The notion of continuity depends upon that of _order_, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry…
Bertrand Russell, Mysticism and Logic and Other Essays
States what Differentiable function is.
But most frequently we consider only the cases in which the functions are such as are called _algebraic_, and to which the idea of _degree_ is applicable. In this case we can give more precision to the general proposition by determining the analytical character which must be necessarily presented by the equation, in order that this property may be verified. It is easy to see, then, that, by the modification just…
Auguste Comte, The philosophy of mathematics
States what Differentiable function is.
Related ideas
- Function (mathematics)
- Real number
- Mathematical analysis
- Complex number
- Continuous function
- Intermediate value theorem
- Weierstrass function
Opposing perspectives
Place in the tradition
Part of Philosophy of mathematics.