More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists.
In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)).
The function f is also be called locally linear at x0 as it is well approximated by a linear function near this point.
Differentiability of real functions of one variable
Differentiability and continuity
See also: Continuous function
If f is differentiable at a point x0, then f must also be continuous at x0.
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.
Most functions that occur in practice have derivatives at all points or at almost every point.
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.
Main article: Smoothness
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
is differentiable at 0, since
However, for x ≠ 0, differentiation rules imply
which has no limit as x → 0.
Continuously differentiable functions are sometimes said to be of class C. A function is of class C 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 if the first k derivatives f′(x), f′′(x), ..., f(x) all exist and are continuous.
If derivatives f exist for all positive integers n, the function is smooth or equivalently, of class C.
Differentiability in higher dimensions
See also: Multivariable calculus
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 x0 and are continuous at the point x0, then the function is differentiable at that point x0.
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 → R defined by
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
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
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.
Differentiable functions on manifolds
If M is a 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. More generally, 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).
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Differentiable function.