See also differentiation and derivative.
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A function (map) is differentiable at some point if it can be well approximated by a linear map near that point. The approximating linear maps at different points together form the derivative of the map.
One may then ask whether the derivative itself is differentiable, and so on. This leads to a hierarchy of ever more differentiable maps, starting with continuous maps and progressing through maps that are times (continuously) differentiable to those that are infinitely differentiable, and finally to those that are analytic. Infinitely differentiable maps are sometimes called smooth.
Differentiability is first defined directly for maps between (open subsets of) a Cartesian space. These differentiable maps can then be used to define the notion of differentiable manifold, and then a more general notion of differentiable map between differentiable manifolds, forming a category called Diff. We have a parallel hierarchy of ever more differentiable manifolds and ever more differentiable maps between them. Since every more differentiable manifold has an underlying less differentiable manifold, we may always consider maps that are less differentiable than the manifolds between which they run.
In all of the following denotes the real numbers and for their -fold Cartesian product. For we write
for the projection map onto the th factor.
When considering convergence of sequences of elements of these sets we regard them as Euclidean metric spaces with the Euclidean norm
The open subsets of the corresponding metric topology are the unions of open balls in .
A function is differentiable if it comes with a function and a function in the positive rational numbers, such that
Given a predicate on the real numbers , let denote the set of all elements in for which holds. A partial function is equivalently a function for any such predicate and set .
A function is differentiable at a subset with injection if it has a function such that for all Archimedean ordered Artinian local -algebras with ring homomorphism and nilradical such that for all , , for all nilpotent elements ,
A function is differentiable at an element if it is differentiable at the singleton subset , and a function is differentiable if it is differentiable at the improper subset of .
(differentiation of real-valued functions on Cartesian space)
Let and let be an open subset.
Then a function is called differentiable at if there exists a linear map such that the following limit exists as approaches zero “from all directions at once”:
This means that for all there exists an open subset containing such that whenever we have .
We say that is differentiable on a subset of if is differentiable at every , and differentiable (tout court) if is differentiable on all of . We say that is continuously differentiable if it is differentiable and is a continuous function.
The map is called the derivative or differential of at .
(Notation and Terminology)
If , as in classical one-variable calculus, then in def. may be identified with a real number, and that number is also called the derivative of at and often written . (In that case, the notation is generally still reserved for the corresponding linear map, with its input denoted by , so that we have .)
(equivalent formulation)
An equivalent way to state def. is to say that
where is a function such that . This is easy to see; just let .
Another equivalent way to say it is that
where are functions such that . For if this is true, then satisfies the previous definition. Conversely, if the previous definition holds, then defining satisfies this definition.
A weaker notion of differentiability is the following:
Let and an open subset.
Then a function is said to have directional derivative in the direction of at if the limit
exists. Here is just a real number.
Historically, the term ‘directional derivative’ was reserved for when is a unit vector (or divide the derivative above by ), but the general concept involves less structure and is more important but has no other established name. If is a standard basis vector , then the directional derivative is called a partial derivative with respect to the corresponding coordinate, and often written or .
If is differentiable at in the sense of def. , then is its directional derivative along . In particular, the coordinates of are the partial derivatives of .
In general, may have all partial derivatives, and even all directional derivatives, without being differentiable; see the examples below. However, if has all partial derivatives and they are continuous as functions of , then in fact is differentiable (and indeed continuously differentiable).
(differentiation of functions between Cartesian spaces)
Let and let be an open subset.
Then a function is differentiable if for all the component function
is differentiable in the sense of def. .
In this case, the derivatives of the assemble into a linear map of the form
For and differentiable manifolds, then a function is called differentiable if for an atlas for and and atlas for then for all and the function
For an open subset and a differentiable function (def. ), we may regard its differential as a function from a subset of (the points where is differentiable) to the space of linear maps. Since is again a Cartesian space, we may then ask whether is differentiable.
We can then iterate, obtaining the following hierarchy of differentiability. Because iterated differentiability by itself is not very useful, and a differentiable map is necessarily continuous, one generally includes continuity of the last assumed derivatives.
The map is continuous or if it is a continuous map between underlying topological spaces. We begin with this since a differentiable map is necessarily continuous.
The map is continuously differentiable or on if it is differentiable at all points of and the resulting map is continuous. (At this point, if we generalize to infinite-dimensional spaces, we get a variety of notions of when the differential is ‘continuous’; see continuously differentiable map for discussion.)
The map is twice differentiable if it is differentiable and its derivative is differentiable. A twice differentiable map must be continuously differentiable. Similarly, is twice continuously differentiable or if it is twice differentiable and the second derivative is continuous.
By recursion, is times differentiable if is times differentiable and the th derivative of is differentiable. Similarly, is times continuously differentiable or if is times differentiable and the th derivative of is continuous.
The map is smooth or infinitely differentiable or if it is times differentiable for all , or equivalently if it is for all . (There is no difference between infinite differentiability and infinite continuous differentiability.) One may also define this notion coinductively: is infinitely differentiable if it is differentiable and its derivative is infinitely differentiable.
One step higher, we may ask whether is analytic or .
Note that is differentiable on a set iff there is a function on (necessarily unique, assuming that has no isolated points) such that
Reverse quantifiers, and is uniformly differentiable on iff there is a function on such that
In classical mathematics, is uniformly differentiable if and only if is differentiable and its derivative is uniformly continuous; in other words, is uniformly differentiable iff is uniformly-continuously differentiable. The same is true in constructive mathematics as long as one assumes dependent choice. In the absence of dependent choice, however, the argument only goes one way, and uniform differentiability is stronger. Furthermore, just as pointwise continuity is not as well behaved constructively as uniform continuity, so pointwise differentiability is not as well behaved constructively as uniform differentiability. For this reason, uniform differentiability is particularly important in constructive mathematics.
In addition, one could talk about locally uniform differentiablilty, which is a continuously differentiable function on a set which is uniformly differentiable on every closed and bounded subset .
If is twice differentiable with , its second derivative
is a function from into the space of bilinear maps from to .
(Symmetry of higher derivatives). If is twice differentiable, then its second derivative lands in the space of symmetric bilinear maps, i.e. for any and we have
It suffices to assume ; otherwise we just consider it componentwise. Define a function by
Then by the chain rule, is differentiable and
where as . Now the mean value theorem tells us that for some we have
But the “second-order difference” is manifestly symmetric in and , so we have
where . But bilinearity of the LHS then implies that it is identically zero.
The components of the bilinear map are the second-order partial derivatives of . Thus, this theorem says that if is twice differentiable, then the mixed partials are equal,
The second-order partial derivatives may exist without the mixed partials being equal; see below for a counterexample. However, the theorem shows that if we require to actually be twice differentiable rather than merely having second-order partial derivatives, then this cannot happen.
In particular, we have the following corollary, which is more commonly found in textbooks.
(Symmetry of continuous higher derivatives). If has first and second-order partial derivatives, and the latter are continuous in a neighborhood of , then the mixed partial derivatives are equal,
Continuity of the second-order partials implies that the first-order partials are differentiable, and hence so is the differential .
Note that the proof of the theorem implies that if is twice differentiable at , then there exists a bilinear map such that
where . This is a condition that makes sense as a condition on an arbitrary without assuming differentiability. and if it holds, then the bilinear map must be symmetric.
Moreover, as explained here, if is differentiable in a neighborhood of and satisfies this condition at , then it is in fact twice differentiable at . To see this, it suffices to show that each coordinate of is differentiable, so let , with a unit basis vector and a real number . Then we have
Taking the limit as and using differentiability of at and (for sufficiently small ), we get
Now take the limit as ; on the left we get by assumption, so the function is differentiable at with derivative . Thus, is differentiable at .
On the other hand, this condition by itself does not even imply that is continuous. For instance, if is a -linear map , then the second-order difference is identically zero.
Let be differentiable. Instead of asking whether is differentiable, we can ask whether its exponential transpose is differentiable. (Note that is the tangent bundle of .) This amounts to asking that for and , we have
for a linear map , where . Setting , we see that this implies that is differentiable, with differential for any . And setting , we obtain for any . Thus, we can write
where is the symmetric bilinear map from the previous section. Conversely, if is twice differentiable, then using linearity and continuity of it is easy to see that the above condition holds.
Therefore, these two kinds of twice-differentiability of are equivalent as conditions on , but the resulting second differential is different. In the second case, we get
rather than merely the first term . There are two advantages to the second approach (asking that be differentiable).
Firstly, we can reformulate it in terms of by asking that there exists a linear map and a bilinear map such that
where . This holds if is twice differentiable, for then we can write
and can also be incorporated into the error term. Of course, setting we obtain the characterization of twice-differentiability from the previous section. But setting , we find that is differentiable at with derivative . So here we have a direct characterization of the second derivative which also implies that the first derivative exists (although it seems that it doesn’t imply differentiability in a neighborhood of , so that the resulting “second derivative” may not actually be the derivative of the first derivative).
Secondly, the virtue of a second differential incorporating the first derivatives is that like the first differential , but unlike the bilinear map , it satisfies Cauchy's invariant rule. This means that we can express the chain rule for second differentials of composite maps simply by substitution: if and , then finding in terms of and , and and in terms of and , and substituting, gives the correct expression for in terms of and .
In fact, this can be proven using the above direct characterization of the second differential, in essentially exactly the same way that we prove the ordinary chain rule for first derivatives. We sketch the proof, omitting the explicit error terms. We can write
as
where and and . Now by extra-strong twice differentiability of , this is approximately equal to
But by differentiability of , we have and , while by extra-strong twice differentiability of we have . Thus, we obtain approximately
which is exactly what we would get by substitution.
If and are -differentiable manifolds, then we may define what it means for a map to be times differentiable, or , for any , by asking that it yield such a map when restricted to any charts. The most common case is when and are smooth (infinitely differentiable) manifolds, so that we can define functions between them for all . (Analytic manifolds, which are necessary in order to define analyticity of , are somewhat rarer.)
A differentiable map between manifolds induces a map between their tangent bundles ; this operation extends to a functor from manifolds and maps to manifolds and maps. See differentiation for more.
The function defined by
is continuous everywhere, and has directional derivatives (def. ) at in all directions, but is not differentiable at in the sense of def. . Note that the (unnormalized) directional derivative along the vector is , which is not linear.
One may wonder whether existence of a linear map is enough, but this is also not the case. The function defined by
has all directional derivatives at equaling , so that in particular there is a linear map whose values are the directional derivatives. But it is not differentiable at ; in fact, it is not even continuous at . (Thus it also provides an example of a discontinuous function which has all directional derivatives.)
The discontinuity kind of gives this last one away; maybe a continuous function with all directional derivatives must be differentiable? But no, because if is a Weierstrass function? (continuous everywhere but differentiable nowhere) from to with period , and is a -argument arctangent function from to (so that and for all and ), then set
Then is, like , continuous everywhere and differentiable nowhere, yet the directional derivatives through the origin are all . There are further examples (including one that is analytic except at the origin) at Math StackExchange question 1497043.
Let be a natural number, and consider the function
from the real line to itself, with . Away from , is smooth (even analytic); but at , is not continuous, is continuous but not differentiable, is differentiable but not continuously differentiable, and so on:
(For a fractional value of , has the same behaviour as .)
Similarly, in two dimensions we can consider functions such as
together with . This is smooth away from , and once differentiable at , even in the strong sense that it is well-approximated by a linear function near . However, its derivative is not continuous at . In particular, this shows that the converse of the theorem “if the partial derivatives exist and are continuous at a point, then the function is differentiable there” fails, even in higher dimensions.
Uniform differentiability is stronger than continuous differentiability but independent of twice differentiability. For an example that is uniformly differentiable but not twice differentiable, use again. For an example that that is twice differentiable (and hence continuously differentiable) but not uniformly differentiable, use . However, on a compact domain, any continuously differentiable function (and a fortiori any twice differentiable function) must be uniformly continuous (at least in classical and intuitionistic mathematics).
The function defined by
plus , has partial derivatives and that both exist but are not equal at (nor are they continuous at ). Therefore, by the theorem proven above, it is not twice differentiable at .
While differentiability means approximability by a linear function, twice differentiability does not mean approximability by a quadratic function. For example, the function
(with , to make it continuous there) is not twice differentiable at , but it is well-approximated by the quadratic polynomial in the sense that their difference is as . Even worse, the function
is not twice differentiable at , but it is well-approximated by a polynomial of any finite degree (namely, the zero polynomial), in the sense that their difference is as .
In some contexts, it is useful to say that functions such as these have “pointwise second derivatives”. More precisely, we say that a function has a pointwise derivative at if it is continuous at and there exists a polynomial of degree such that
(We have to state continuity explicitly in order to match the usual notion of differentiability when , since nothing in this limit constrains the value of .) In this case, the pointwise derivative is (and it follows that all lower-order pointwise derivatives match as well). Thus we would say that while does not have a second derivative at , it does have a pointwise second derivative at , and . Similarly, is pointwise smooth. See e.g. this MSE answer.
The polynomial may be thought of as the th-order Taylor polynomial of at , and the limit above is Taylor’s theorem with the Peano remainder. (Thus the subscript can be thought of as standing for ‘Peano–Taylor’ as well as for ‘pointwise’.) Note that while -times pointwise differentiability is weaker than -times differentiability for and equivalent for , it is stronger for (since even -times pointwise differentiability requires continuity).
In the definition of strong twice-differentiability, we cannot replace the symmetric bilinear map by the corresponding quadratic form . In other words, if we suppose that
where , for some quadratic form , it does not follow that is twice differentiable at , even in one dimension. The same old counterexample
(with ) works: we have
so .
continuous function, differentiable function, continuously differentiable function, smooth function, analytic function
Early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:
Last revised on December 1, 2023 at 20:14:06. See the history of this page for a list of all contributions to it.